# Probability and Mathematical Physics Seminar

## Spring Semester 2014

The seminar covers a wide range of topics in pure and applied probability and in mathematical physics.
The seminar is run by Chuck Newman, S. R. Srinivasa Varadhan, Gérard Ben Arous and Shirshendu Chatterjee.

Usual place and time are Warren Weaver Hall room 512 on Fridays at 11:00 AM - 12:00 noon, but check each announcement since this is sometimes changed.

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Wednesday, January 29 - First seminar of a seminar series

Professor Charles Newman, New York University

Statistical Mechanics and the Riemann Hypothesis.

Abstract:
Find the details here.

Warren Weaver Hall Room 1314 from 2:00 PM to 3:50 PM.

Friday, January 31

Ivan Matic, Baruch Collegem City University of New York

A sublinear bound for the variance of solutions for Hamilton-Jacobi equations.

Abstract:
We study the solutions to random Hamilton-Jacobi equations. The Hamiltonian is assumed to be stationary and ergodic with respect to translations. In dimensions two and higher we will prove that as the time t increases, the variance of the solution increases at a rate slower than t/log(t).

Warren Weaver Hall Room 512 at 11:00 AM.

Wednesday, February 5 - Second seminar of a seminar series

Professor Charles Newman, New York University

Statistical Mechanics and the Riemann Hypothesis.

Abstract:
Find the details here.

Warren Weaver Hall Room 1314 from 2:00 PM to 3:50 PM.

Friday, February 7

Martin Tassy, Brown University

Translation invariant Gibbs measures on tilings by Nx1 Rectangles.

Abstract:
For the classic case of 2x1 dominos the set of ergodic gibbs measure on tilings is well understood and can be characterized by 2 parameters known as the horizontal and vertical slope of the measure. We will show how this result can be extended a in surprising way for Nx1 rectangle tilings through the conway tiling group. And we will explain why the approach used to obtain our results should be relevant for a large class of non-integrable tilings.

Warren Weaver Hall Room 512 at 11:00 AM.

Wednesday, February 12 - Third seminar of a seminar series

Professor Charles Newman, New York University

Statistical Mechanics and the Riemann Hypothesis.

Abstract:
Find the details here.

Warren Weaver Hall Room 1314 from 2:00 PM to 3:50 PM.

Friday, February 14

Herbert Spohn, IAS Princeton and TUM Munich

The noisy Burgers equation and interacting diffusions.

Abstract:
In its spatially integrated version the noisy Burgers equation becomes the one-dimensional Kardar-Parisi-Zhang equation, on which there has been a lot of activities in the recent years. I will discuss two models which are expected to have the same fluctuation statistics as the Burgers equation. (1) The sound mode of a Hamiltonian particle system and (2) nonreversible interacting diffusions, in particular with point interactions.

Warren Weaver Hall Room 512 at 11:00 AM.

Wednesday, February 19 - Forth seminar of a seminar series

Professor Charles Newman, New York University

Statistical Mechanics and the Riemann Hypothesis.

Abstract:
Find the details here.

Warren Weaver Hall Room 1314 from 2:00 PM to 3:50 PM.

Friday, February 21

Alexander Fribergh, Universit de Toulouse

Biased random walk on supercritical percolation clusters.

Abstract:
We will present results on biased random walks on supercritical percolation clusters. This is a natural model for observing trapping phenomena and anomalous long-term behaviors. We will explain why this model exhibits a phase transition from positive speed to zero speed as the bias increases. Furthermore, we shall discuss a subtle difficulty appearing when trying to rescale such a process to obtain scaling limits.
This talk will be based on past and ongoing work of Alexander Fribergh and Alan Hammond.

Warren Weaver Hall Room 512 at 11:00 AM.

Wednesday, February 26 - Fifth seminar of a seminar series

Professor Charles Newman, New York University

Statistical Mechanics and the Riemann Hypothesis.

Abstract:
Find the details here.

Warren Weaver Hall Room 1314 from 2:00 PM to 3:50 PM.

Friday, February 28 - No Seminar

Minerva Lecture at Columbia University (Room 903, 1255 Amsterdam Ave)

Wednesday, March 5 - Sixth seminar of a seminar series

Professor Charles Newman, New York University

Statistical Mechanics and the Riemann Hypothesis.

Abstract:
Find the details here.

Warren Weaver Hall Room 1314 from 2:00 PM to 3:50 PM.

Friday, March 7

Nicola Kistler, College of Staten Island, CUNY

A multi-scale refinement of the 2nd moment method.

Abstract:
A fundamental tool in the study of the extremes of large combinatorial structures is the so-called 2nd moment method. In case of correlated random fields, however, a plain application is (more often than not) inconclusive. I will discuss the main steps behind a refinement of the method which is seemingly very effective whenever multiple scales can be identified.

Warren Weaver Hall Room 512 at 11:00 AM.

Wednesday, March 12 - Seventh and last seminar of a seminar series

Professor Charles Newman, New York University

Statistical Mechanics and the Riemann Hypothesis.

Abstract:
Find the details here.

Warren Weaver Hall Room 1314 from 2:00 PM to 3:50 PM.

Friday, March 14 - Two Consecutive Talks

Oren Louidor, The Technion (Israel)

The thinned extremal process of the 2D discrete Gaussian Free Field.

Abstract:
We consider the discrete Gaussian Free Field in a square box of side N in Z2 with zero boundary conditions and study the joint law of its extreme values (h) and their spatial positions (x), properly centered and scaled. Restricting attention to extreme values which are also local maxima in a neighborhood of radius rN, we show that when N, rN → ∞ with rN/N → 0, the joint law above converges weakly to a Poisson Point Process with intensity measure Z(dx) e-α h dh, where α =  2π  and Z(dx) is a random measure on [0,1]2. In particular, this yields an integral representation for the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. Time permitting, I will discuss various properties of the Z measure, including conformal covariance and star-scale-type invariance, similar to that found in Gaussian multiplicative chaos. Joint work with Marek Biskup (UCLA).

Warren Weaver Hall Room 512 at 10:00 AM.

Local quantum unique ergodicity for random matrices.

Abstract:
For generalized Wigner matrices, I will explain a probabilistic version of quantum unique ergodicity at any scale, and gaussianity of the eigenvectors entries. The proof relies on analyzing the effect of the Dyson Brownian motion on eigenstates. Relaxation to equilibrium of the eigenvectors is related to a new multi-particle random walk in a random environment, the eigenvector moment flow. This is joint work with H.-T. Yau.

Warren Weaver Hall Room 512 at 11:00 AM.

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Friday, March 21 - No Seminar

Spring Break

TBA.

Abstract:
TBA

Warren Weaver Hall Room 512 at 11:00 AM.

Friday, April 4 - Two Consecutive Talks

Christophe Garban, CNRS ENS Lyon, UMPA

Liouville Brownian motion.

Abstract:
Let X be a Gaussian Free Field (GFF) in two dimensions. I will introduce a Feller process (PtX) on the plane which, a.s. in the realization of the GFF X, preserves the so-called Liouville measure defined formally by "M(dx)=eγXdx" (with γ < γc=2). The Liouville measure was popularized a few years ago by Duplantier and Sheffield in the context of Liouville Quantum gravity. I will discuss the construction and the properties of this Feller process called the Liouville Brownian motion as well as some recent progresses on the super-critical Liouville Brownian motion, i.e. when γ > γc=2.
This is based on joint works with N. Berestycki, R. Rhodes, and V. Vargas.

Warren Weaver Hall Room 512 at 10:00 AM.

J. Theodore Cox, Syracuse University

Convergence of finite voter model densities.

Abstract:
See the abstract here.

Warren Weaver Hall Room 512 at 11:00 AM.

Friday, April 11 - No Seminar

Columbia-Princeton Probability Day 2014

Friday, April 18 - Two Consecutive Talks

Claude Godrche, Institut de Physique Thorique, Saclay

Dynamics of Ising spin systems.

Abstract:

The best introduction to this talk is the following quotation of Glauber in his celebrated 1963 article: “The principles of nonequilibrium statistical mechanics remain in largest measure unformulated. While this lack persists, it may be useful to have in hand whatever precise statements can be made about the time-dependent behavior of statistical systems, however simple they may be. We have attempted, therefore, to devise a form of the Ising model whose behavior can be followed exactly, in statistical terms, as a function of time. While certain of the assumptions underlying the model are to a degree arbitrary, it is surely one of the simplest ones involving N coupled particles for which exact time-dependent solutions can be found”.

The aim of this talk is to illustrate some facets of the progress achieved since Glauber on his model:

(i) The interplay between reversibility and Gibbsianity

(ii) A glimpse at the dynamics

Warren Weaver Hall Room 512 at 10:00 AM.

Lionel Levine, Cornell University

Sandpiles and system-spanning avalanches.

Abstract:
A sandpile on a graph is an integer-valued function on the vertices. It evolves according to local moves called topplings. Some sandpiles stabilize after a finite number of topplings, while others topple forever. For any sandpile S0 if we repeatedly add a grain of sand at an independent random vertex, we eventually reach a sandpile Sτ that topples forever. Statistical physicists Poghosyan, Poghosyan, Priezzhev and Ruelle conjectured a precise value for the expected amount of sand in this “threshold state” Sτ in the limit as S0 goes to negative infinity. I will outline the proof of this conjecture in the paper and explain the big-picture motivation, which is to give more predictive power to the theory of “self-organized criticality”.

Warren Weaver Hall Room 512 at 11:00 AM.

Friday, April 25

Jean Bertoin, Universitt Zrich

Fragmentation-dilation processes and their limits.

Abstract:
A new class of fragmentation type processes is introduced, in which the accumulation of small dislocations which would instantaneously shatter the entire mass into dust, is compensated by an adequate dilation of fragments. A main feature of these processes is that their evolution is govern by a dislocation measure which can be much more general than usual.

Warren Weaver Hall Room 512 at 10:00 AM.

Friday, May 2

Krishnamurthi Ravishankar, SUNY New Paltz.

Convergence to critical measure in one dimensional Zero range Processes with ergodic site-wise disorder.

Abstract:
Find the abstract here.

Warren Weaver Hall Room 512 at 11:00 AM.

Friday, May 9

Jon Peterson, Purdue University

Recurrence, transience, and ballisticity for one-dimensional excited random walks with bounded jumps.

Abstract:
Excited random walks (also called cookie random walks) are a model for a self-interacting random motion where the transition probabilities are a function of the local time of the random walk. Much is known about one-dimensional excited random walks including a criterion for recurrence/transience and a criterion for ballisticity of the walk. For the model that is usually studied, the random walk always moves in steps of size one. In this talk we generalize the model of excited random walks to include bounded jumps. There are some natural generalizations of the results from the nearest neighbor case, but the proofs of some of the facts (such as the criterion for ballisticity) completely break down when the walk has bounded jumps. Nevertheless, in certain cases we are able to use completely different methods to prove the natural conjectures for recurrence/transience and ballisticity.
This is joint work with Burgess Davis.

Warren Weaver Hall Room 512 at 11:00 AM.

Friday, May 16

Lingjiong Zhu, University of Minnesota.

Phase Transitions in Directed Graphs.

Abstract:
Phase transitions for exponential random graphs have been studied rigorously by Chatterjee-Diaconis and Radin-Yin. For a family of directed exponential random graph models parametrized by edges and outward stars, we study the phase transitions as well as the critical exponent along the phase transition curve and at the critical point. A related constrained graph model will also be discussed. This is based on recent joint work with David Aristoff..

Warren Weaver Hall Room 512 at 11:00 AM.

## Fall Semester 2013

Friday, September 6 - Two Consecutive Talks

Jing Wang, Purdue University

The subelliptic heat kernel and small time asymptotics on sub Riemannian model spaces.

Abstract:

We work on model spaces of sub Riemannian manifolds: the Cauchy-Riemann sphere S2n+1, the CR complex hyperbolic space H2n+1 and the Quaternionic sphere S4n+3. On each space there is a canonical di usion operator L: The sub-Laplacian, which is not elliptic but only subelliptic.

The symmetries of these model spaces enable us to obtain an explicit and geometrically meaningful formula for each associated heat kernel. From them we can deduce the small-time behaviors of the heat kernels on the diagonal, on the cut-locus, and outside of the cut-locus. <\p>

The key point is to work in cylindrical coordinates that reect the symmetries coming from the Hopf bration of these model spaces. This is a joint work with F. Baudoin.

Warren Weaver Hall Room 512 at 10:00 AM.

Amir Dembo, Stanford University

Persistence Probabilities.

Abstract:

Persistence probabilities concern how likely it is that a stochastic process has a long excursion above fixed level and of what are the relevant scenarios for this behavior. Power law decay is expected in many cases of physical significance and the issue is to determine its power exponent parameter. I will survey recent progress in this direction (jointly with Sumit Mukherjee), dealing with stationary Gaussian processes that arise from random algebraic polynomials of independent coefficients and from the solution to heat equation initiated by white noise.

If time permits, I will also discuss the relation to joint works with Jian Ding and Fuchang Gao, about persistence for iterated partial sums and other auto-regressive sequences, and to the work of Sakagawa on persistence probabilities for the height of certain dynamical random interface models.

Warren Weaver Hall Room 512 at 11:00 AM.

Friday, September 13 - Two Consecutive Talks

Antonio Auffinger, University of Chicago

On some properties of mean field spin glasses.

Abstract:
Spin glasses are magnetic systems exhibiting both quenched disorder and frustration, and have often been cited as examples of "complex systems." As mathematical objects, they provide several fascinating structures and conjectures. In this talk, we overview some recent progress in mean field models that include the famous Sherrigton-Kirkpatrick model and the bipartite model. We will focus on properties of the energy landscape and of the functional order parameter. We will explain how these properties help to shed more light in the mysterious and beautiful solution proposed 30 years ago by G. Parisi. Based on joint works with Wei-Kuo Chen.

Warren Weaver Hall Room 512 at 10:00 AM.

Ofer Zeitouni, Weizmann Institute of Science and NYU

Performance of the Metropolis algorithm on a disordered tree: the Einstein relation.

Abstract:
Consider a d-ary rooted tree (d ≥ 3) where each edge e is assigned an i.i.d. (bounded) random variable X(e) of negative mean. Assign to each vertex v the sum S(v) of X(e) over all edges connecting v to the root, and assume that the maximum Sn* of S(v) over all vertices v at distance n from the root tends to infinity (necessarily, linearly) as n tends to infinity. We analyze the Metropolis algorithm on the tree and show that under these assumptions there always exists a temperature 1⁄β of the algorithm so that it achieves a linear (positive) growth rate in linear time. This confirms a conjecture of Aldous (Algorithmica, 22(4):388-412, 1998). The proof is obtained by establishing an Einstein relation for the Metropolis algorithm on the tree. (Joint work with Pascal Maillard arXiv)

Warren Weaver Hall Room 512 at 11:00 AM.

Friday, September 20

Alon Nishry, Princeton University

Hole probability for entire functions represented by Gaussian Taylor series.

Abstract:
We study the hole probability of Gaussian entire functions. More specifically, we work with entire functions given by a Taylor series with i.i.d. complex Gaussian random variables and arbitrary non-random coefficients. A 'hole' is the event where the function has no zeros in a disk of radius r centered at 0. We find exact logarithmic asymptotics for the rate of decay of the hole probability for large values of r, outside a small (non-random) exceptional set.

Warren Weaver Hall Room 512 at 11:00 AM.

Friday, September 27 - No Seminar

Courant Instructor Day!

Friday, October 4 - No Seminar at the Courant Institute

Columbia / Courant Joint Probability Seminar Series at Columbia University

Kardar-Parisi-Zhang Universality

Invited speakers are:

Herbert Spohn (TU Munich)
Jeremy Quastel (University of Toronto)
Leonid Petrov (Northeastern University)

Wednesday, October 9 - Note different time and place

Couplings and attractiveness for interacting particle systems.

Abstract:

I will speak of joint works with Thierry Gobron (CNRS, Cergy-Pontoise) and Lucie Fajfrova (UTIA, Prague).

Attractiveness for particle systems corresponds to the existence of a coupling of two processes with the same infinitesimal generator, that stay ordered as soon as it is the case for their initial states. I will mainly focus on generalized misanthrope models. They are conservative particle systems on Zd for which the “basic coupling” construction is not possible under necessary and sufficient conditions for attractiveness. For such models, in each transition, k particles may jump from a site x to another site y, with k ≥ 1, and the jump rate depends on the number of particles only at sites x and y. Under attractiveness conditions, I will explain the increasing coupling we have constructed, and how it permits to determine the extremal invariant and translation invariant measures for the dynamics. I will present examples of generalized zero-range, generalized target and generalized misanthrope models. Finally I will explain how to deal with attractiveness for exclusion processes with speed change, and for non-conservative dynamics.

Warren Weaver Hall Room 517 at 11:00 AM.

Friday, October 11 - Two consecutive talks

Charles Radin, University of Texas, Austin

Phase Diagrams of Large Graphs.

Abstract:
Our goal is to understand the phase diagrams of large graphs which are subject to variable structural constraints, as a natural extension of extremal graph theory. Our approach uses a variational principle on a space of graphons.

Warren Weaver Hall Room 512 at 10:00 AM.

Mark Rudelson, University of Michigan

Delocalization of eigenvectors of random matrices with independent entries.

Abstract:
Let A be an n by n random matrix with independent centered entries having exponential type tail decay and unit variances. We prove that, with high probability, the eigenvalues of A are delocalized, i.e., all coordinates of any unit eigenvector have the magnitude O(n^{-1/2}) up to logarithmic terms. Joint work with Roman Vershynin.

Warren Weaver Hall Room 512 at 11:00 AM.

Friday, October 18

Sanchayan Sen, Courant Institute

The structure of critical inhomogeneous random graphs.

Abstract:

Consider the rank-1 inhomogeneous random graph on n vertices constructed from i.i.d. positive random variables W1,...,Wn by placing an edge between i and j with probability (1 - exp( - Wi Wj / ∑k Wk )) independently for each i ≠ j. The model is critical if E( W1) = E((W1)2). We show that after assigning mass Wi / n-2/3 to vertex i and scaling the graph distance by n-1/3, the components viewed as measured metric spaces converge in Gromov-Hausdorff-Prokhorov topology to some limiting (random) compact, measured metric spaces if W1 satisfies appropriate conditions.

Joint work with Shankar Bhamidi, Amarjit Budhiraja and Xuan Wang.

Warren Weaver Hall Room 512 at 11:00 AM.

Friday, October 25

Jinho Baik, University of Michigan

Random matrix with locally-varying potential.

Abstract:
The universality of the sine kernel for unitary ensembles with external potential is well-known. Here the relevant potentials are those which are flat in the scale of the mean spacing of the eigenvalues. We consider what happens if the potential varies rapidly in the scale of the mean spacing of the eigenvalues. We evaluate the correlation functions for the circular unitary ensembles with locally-varying potentials and show how the sine kernel is modified due to the potential.

Warren Weaver Hall Room 512 at 11:00 AM.

Friday, November 1

Ioannis Karatzas, Columbia University

Competing Brownian Particle Systems.

Abstract:
See the abstract here.

Warren Weaver Hall Room 512 at 11:00 AM.

Friday, November 8 - Two Consecutive talks

Michael Marcus, The City University of New York

Intersection local times, loop soups and permanental Wick powers.

Abstract:
Several stochastic processes related to transient Lévy processes with potential densities that need not be symmetric nor bounded on the diagonal, are defined and studied. The processes include n-fold self-intersection local times of transient Léevy processes and permanental chaoses, which are “loop soup n-fold self-intersection local times” constructed from the loop soup of the Léevy process. Loop soups are also used to define permanental Wick powers, which generalize standard Wick powers, a class of n-th order Gaussian chaoses. Dynkin type isomorphism theorems are obtained that relate the various processes.

Warren Weaver Hall Room 512 at 10:00 AM.

Shirshendu Chatterjee , Courant Institute.

The order-chaos phase transition for a general class of complex Boolean networks.

Abstract:
We consider a model for heterogeneous gene regulatory networks that is an “annealed approximation” of Kauffmann's (1069) original random Boolean networks. In this model, genes are represented by the nodes of a random directed graph Gn on n vertices with specified in-degree distribution pin (resp. out-degree distribution pout or joint distribution pin,out of in-degree and out-degree), and the interactions (through certain Boolean functions) among the genes are approximated by an appropriate threshold contact process (in which a vertex with at least one occupied in-neighbor at time t will be occupied at time t+1 with probability q, and vacant otherwise) on Gn. We characterize the order-chaos phase transition curve segregating the chaotic and ordered random Boolean networks.
The talk is based on a recent work, available at the arXiv.

Warren Weaver Hall Room 512 at 11:00 AM.

Friday, November 15 - Two Consecutive talks

Jonathan Mattingly , Duke University

Uniqueness of the inviscid limit in a simple model damped/driven system.

Abstract:

I will talk about the problem of understanding the long time dynamics of a damped driven system when the damping (and possibly the driving) tend to zero. I will begin by reviewing some motivation from inviscid limits of fluid type equations. I will talk about the difference between systems which develop anomalous dissipation and those which don't.

However, I will spend most of the time talking about a simple 3-dimensional SDE which is a toy for a model truncation of a "fluid" system. I will discuss in detail how to show this model system has a unique limiting measure which is selected as the damping and stochastic driving is taken to zero. This uniqueness holds even though the relative limiting martingale problem does not have a unique solution.
The bulk of the talk will be based on a joint work with Etienne Pardoux.

Warren Weaver Hall Room 512 at 10:00 AM.

Stefano Olla, Université Paris Dauphine

Diffusion and super-diffusion of energy in one dimensional systems of oscillators.

Abstract:
We consider a system of harmonic oscillators with stochastic perturbations of the dynamics that conserve energy and momentum. In the one dimensional unpinned case, under proper space-time rescaling, we prove that Wigner distribution of energy converges to the solution of a fractional heat equation (with power 3/4 for the laplacian). For pinned systems or in dimension 3 or higher, we prove normal diffusive behaviour. Similar results are also obtained for space-time energy correlations in equilibrium. This is in agreement with previous 'weak noise' limits, passing through a kinetic equation, and conjectured behaviour for beta-FPU chains (quartic symmetric interaction).
Joint works with Tomasz Komorowski and Giada Basile.

Warren Weaver Hall Room 512 at 11:00 AM.

Friday, November 22 - No Seminar

Northeast Probability Seminar at CUNY

Invited speakers are

Martin Hairer (University of Warwick)
Marius Junge (UIUC)
Ashkan Nikeghbali (University of Zurich)
Ruth Williams (UCSD)

Friday, November 29 - No Seminar

Thanksgiving Holidays!

Wednesday, December 4 - Note that the time is different from the usual

Strong solutions of infinite-dimensional stochastic differential equations.

Abstract:
Interacting Brownian motions (IBMs) in infinite-dimensions are infinitely many Brownian particles moving in Rd with interaction potentials. We give general theorems to construct unique, strong solutions of infinite-dimensional stochastic differential equations describing IBMs in infinite-dimensions. As applications, we construct infinite-dimensional labeled dynamics arising from random matrices. In fact, the associated unlabeled dynamics are reversible with respect to random point fields related to random matrices. Typical examples are Dyson, Airy, and Bessel random point fields, and the Ginibre random point field. All canonical Gibbs measures with Ruelle's class interaction potentials (satisfying suitable marginal assumptions) are covered by our theorems. In particular, we detect the infinite-dimensional stochastic differential equations describing the stochastic dynamics related to Airy random point fields with β = 1,2,4. When β = 2, this dynamics coincides with that given by the space-time correlation functions constructed by Spohn, Johansson, and others. This is joint work with Hideki Tanemura.

Warren Weaver Hall Room 512 at 11:00 AM.

Friday, December 6 - Two Consecutive Talks

Francesco Russo, ENSTA ParisTech

Stochastic calculus via regularization in Banach spaces: path dependent calculus and stochastic evolution equations.

Abstract:
See the abstract here.

Warren Weaver Hall Room 512 at 10:00 AM.

Dynamical rigidity of Ginibre interacting Brownian motions.

Abstract:
Ginibre interacting Brownian motions in infinite-dimensions are infinitely many Brownian particles in R2 interacting via the 2-dimensional Coulomb potential with inverse temperature β = 2. The invariant probability measure of the associated unlabeled dynamics is the Ginibre random point field. It is known that the Ginibre random point field has various (geometric) rigidities. In this talk, as a dynamical counter part, Ginibre interacting Brownian motions in infinite-dimensions have dynamical rigidities totally different from interacting Brownian motions with Ruelle's class potentials.

Warren Weaver Hall Room 512 at 11:00 AM.

## Spring Semester 2013

Friday, February 1

Hana Kogan, Courant Institute
Zero temperature Stochastic Ising Model on the slabs in Z^3.
Abstract:
We consider zero temperature Glauber dynamics on the vertices of the the graph Z^2 x k for integer k >1. This is a slab of thickness k from Z^3. At time 0 each vertex is assigned value 1 or -1 with equal probability independent of all other vertices. The system is updated when a clock with exponentially distributed waiting time rings at a vertex. The spin at this vertex then changes its sign to agree with the majority of its neighbors. If there is a tie, it flips with probability 1/2 independent of everything else. We say that a site fixates if it changes its value finitely many times. Results: A site in the slab of any thickness fixates with positive probability. A site in the slab of thickness 2 and 3 fixates with probability 1. A site in the slab of thickness 5 or greater does NOT fixate with positive probability. Conjectures: A site in the slab of thickness 4 fixates with probability 1 Let R(t) be the maximal radius of the same sign cluster at site x. Then R(t) is a bounded function of time a.s. Joint work in progress with C.M.Newman, M.Damron, V.Sidoravicius.
Warren Weaver Hall Room 512 at 11:00 am.

Friday, February 8

Elena Kosygina, Baruch College, City University of New York
Crossing speeds of random walks among sparse'' or spiky'' Bernoulli potentials on integers.
Abstract:
We consider a random walk among i.i.d. obstacles on Z under the condition that the walk starts from the origin and reaches a remote location y. The obstacles are represented by a killing potential, which takes value M>0 with probability p and value 0 with probability 1-p, 0 < p < 1, independently at each site of the lattice. We consider the walk under both quenched and annealed measures. It is known that under either measure the crossing time from 0 to y of such walk, H(y), grows linearly in y. More precisely, the expectation of H(y)/y converges to a limit as y goes to infinity. The reciprocal of this limit is called the asymptotic speed of the conditioned walk. We study the behavior of the asymptotic speed in two regimes: (1) as p goes to 0 for M fixed (sparse''), and (2) as M goes to infinity for p fixed (spiky''). We observe and quantify a dramatic difference between the quenched and annealed settings.
Warren Weaver Hall Room 512 at 11:00 am.

Friday, February 15

Partha Dey, Courant Institute.
Energy Landscape for large average' Gaussian submatrices.
Abstract:
The problem of finding large average submatrices of a real-valued matrix arises in the exploratory analysis of data from a variety of disciplines, ranging from genomics to social sciences. We provide a detailed asymptotic analysis of large average submatrices of an n × n Gaussian random matrix. For k << log n we identify the average and the joint distribution of the k × k submatrix with largest average value. As a dual result, we establish that, for any given τ > 0, the size of the largest square sub-matrix with average bigger than τ is, for large n, equal to one of two consecutive integers near 4(log n - log log n)/τ2.
We then turn our attention to submatrices with dominant row and column sums, which arise as the local optima of iterative search procedures for large average submatrices. For fixed k, we identify the average and joint distribution of a typical k × k submatrices with dominant row and column sums, and we carry out a detailed analysis of the number Ln(k) of such submatrices, beginning with the mean and variance of Ln(k) which has a very atypical behavior. In particular, for k = 2 and k = 3, the order of the means are o(n2) and o(n3), while the variances are n8/3 and n9/2, respectively, with logarithmic corrections. Our principal result is a Gaussian central limit theorem for Ln(k) based on a new variant of Stein's method, that is of independent interest. Based on joint work with Shankar Bhamidi and Andrew Nobel.
Warren Weaver Hall Room 512 at 11:00 am.

Friday, February 22

Jay Rosen, College of Staten Island, City University of New York
Markovian loop soups, permanental processes and isomorphism theorems.
Abstract:
We show how to construct loop soups for general Markov processes and explain how loop soups offer a deep understanding of Dynkin's isomorphism theorem, and beyond.
Warren Weaver Hall Room 512 at 11:00 am.

Wednesday, March 6 - Note the different day

Kshitij Khare , University of Florida
Convergence for some multivariate Markov chains with polynomial eigenfunctions.
Abstract:
In this talk, we will present examples of multivariate Markov chains for which the eigenfunctions turn out to be well-known orthogonal polynomials. This knowledge can be used to come up with exact rates of convergence for these Markov chains. The examples include the multivariate normal autoregressive process and simple models in population genetics. Then we will consider some generalizations of the above Markov chains for which the stationary distribution is completely unknown. We derive upper bounds for the total variation distance to stationarity by developing coupling techniques for multivariate state spaces. The talk is based on joint works with Hua Zhou and Nabanita Mukherjee.
Warren Weaver Hall Room 512 at 11:00 am.

Friday, March 8 - Two consecutive talks

Dmitry Jakobson, McGill University.
Gaussian measures on manifolds of Riemannian metrics.
Abstract:
We first discuss joint work with I. Wigman and Y. Canzani, where we construct a Gaussian measure on a conformal class of Riemannian metrics on a compact surface, centered at a reference metric g_0 with non-vanishing Gauss curvature. We estimate the probability that a random conformal deformation of g_0 will change the sign of the curvature; we compare that probability for different reference metrics g_0. Related questions are then considered for scalar curvature in higher dimensions, as well as for Q-curvature. Next, we shall speak about joint work in progress with B. Clarke, N. Kamran, L. Silberman and J. Taylor. We define Gaussian measures on manifolds of metrics with the ﬁxed volume form. We next compute the moment generating function for the L^2 (Ebin) distance to the reference metric. As time permits, we shall also outline some ideas in a recent work with L. Chen, where we generalize the results of Duplantier and Sheffield in dimension 2 to four dimensions. Duplantier and Sheffield used the 2D Gaussian free field to construct the Liouville quantum gravity measure on a planar domain, and gave a probabilistic proof of the KPZ relation in that setting. We have applied a similar approach to generalize part of their results to R^4: we construct a random Borel measure on R^4 which formally has the density (with respect to the Lebesgue measure) given by the exponential of an instance of the 4D Gaussian free field. We also establish the KPZ relation corresponding to this random measure.
Warren Weaver Hall Room 512 at 10:00 am.

Charles Bordenave, Université de Toulouse.
Large deviations for Wigner matrices without gaussian tails.
Abstract:
We consider a Wigner matrix : a random Hermitian matrix X of size n whose entries above the diagonal are independent and identically distributed with unit variance. Since the seminal work of Wigner in the 50's, it is known that the empirical distribution of the eigenvalues of X / sqrt n converges to the semi-circular law. In 1997, Ben Arous and Guionnet have established a large deviation principle (LDP) around the semi-circular law when the entries are Gaussian. The associated rate function is the Voiculescu's non-commutative entropy. Their proof was based on the explicit formula for the joint law of the eigenvalues, and deyond this result, establishing LDP's for Wigner matrices remains largely open. When the entries are of Weibull type but not subgaussian (for example exponential) we will see that it is however possible to prove such LDP using ideas coming from random graphs. This is a joint work with Pietro Caputo (Univ. Roma Tre).
Warren Weaver Hall Room 512 at 11:00 am.

Friday, March 15

Julien Dubédat, Columbia University
Double dimers and tau-functions
Abstract:
The double dimer model is a variation of the classical dimer model consisting in superimposing two independent dimer configurations (perfect matchings) on a graph, thus creating an ensemble of non-intersecting loops. Kenyon has recently introduced and studied anyonic" correlators for this model. We discuss the convergence (in the small mesh limit) of some of these correlators to the tau-functions appearing in the theory of isomonodromic (SU(2)) deformations.
Warren Weaver Hall Room 512 at 11:00 am.

Friday, March 22 - No Seminar

Spring Break!

Friday, March 29 - No Seminar

Columbia-Princeton Probability day. See the website http://orfe.princeton.edu/conferences/cp13/

Friday, April 5

Matan Harel, Courant Institute
Localization in Random Geometric Graphs with Too Many Edges.
Abstract:
Consider a random geometric graph G(n, r), given by taking a Poisson Point Process of intensity n on the d-dimensional unit torus and connecting any two points whose distance is smaller than r. We condition this model on the rare event that the observed number of edges |E| exceeds its expected value mu by a multiplicative constant larger than one - i.e. greater than (1 + delta) mu, for some fixed positive delta. We prove that, with high probability, this implies the existence of a ball of diameter r with approximately sqrt{2 delta mu} vertices, making up a clique with all the "extra" edges in the graph.
Warren Weaver Hall Room 512 at 11:00 am.

Friday, April 12

Trees in random sparse graphs with given degree sequence.
Abstract:
Let G^D be the set of graphs G(V, E), |V| = n, with degree sequence equal to D = (d_1, d_2, . . . , d_n). What does a graph look like when it is chosen uniformly out of G^D ? This has been studied when G is a dense graph ,|E| = O(n^2), in the sense of graphons or when G is very sparse, (d_n)^2 = o(|E|). We investigate this question in the case of sparse graphs with almost given degree sequence, and give the finite tree subgraph structure of G, under some mild conditions. For graphs with given degree sequence, we re-derive the tree structure in dense and very sparse case to give a continuous picture. Moreover, we are able to show the result for general bipartite graphs with given degree sequence without any further conditions.
Warren Weaver Hall Room 512 at 11:00 am.

Wednesday, April 17 - - Two non-consecutive talks, Note the time and room no

Jayadev Athreya , University of Illinois at Urbana-Champagne
Gap Distributions and Homogeneous Dynamics.
Abstract:
We describe the study of gap distributions for various sequences arising in number theory, geometry, and dynamical systems. The examples we describe are unified by a use of homogeneous dynamics (actions of subgroups of a Lie group) on parameter spaces of geometric objects. The talk will be accessible to a general audience, and will be of a survey nature. Parts of this talk are based on joint work with Y. Cheung, and parts on joint work with J. Chaika, and S. Lelievere.
Warren Weaver Hall Room 512 at 11:00 am.

Brendan Farrell, California Institute of Technology
Random Subspaces in High Dimensions.
Abstract:
We address the angles between two random subspaces. As fundamental as random subspaces are, relevant results in both random matrix theory and free probability have previously been limited to uniformly distributed subspaces. We present the first universality result in this area of random matrix theory, namely for the Jacobi ensemble or MANOVA matrices. We also generalize a fundamental theorem of Voiculescu relating free probability and random matrices to a family of unitary matrices with structure and relatively little randomness. Both approaches provide a connection to discrete harmonic analysis. This is partially joint work with Laszlo Erdos and Greg Anderson.
Warren Weaver Hall Room 1314 at 3:30pm to 5pm.

Friday, April 19

Krishnamurthi Ravishankar, SUNY New Paltz.
Voter Model Perturbation on Z and Brownian Net with Killing.
Abstract:
I will start with the description of a modification of the nearest neighbor voter model (with possibly more than 2 opinions or colors) where in addition to the voter dynamics the colors switch at random in the bulk and at the boundaries. To specify the colors at time t > 0 one follows the dual (genealogy) process which is coalescing random walk with branching and killing until either killing point or time zero is reached and then move up along the arrows with color information to obtain the color at time t. The diffusive scaling limit of the dual where the branching and killing are taken to zero at the appropriate rate is a Brownian net with killing. I will describe the construction and some of the relevant properties of this object. These results apply to Potts model and its continuum limit and are the one dimensional counterpart to recent results of Cox, Durrett and Perkins in three or more dimensions.
(The first part is joint work with C.M. Newman and Y. Moylevskky and the second part is joint work with C.M. Newman and E. Schertzer.)
Warren Weaver Hall Room 512 at 11:00 am.

Wednesday, April 24

Arjun Krishnan, Courant Institute.
Stochastic Homogenization on the Lattice: a Variational Formula for First-Passage Percolation.
Abstract:
Consider a set of positive edge-weights {τe}e ∈ Zd that are stationary and ergodic under translation on the square lattice Zd. Let T(x) be the first-passage time from the origin to x ∈ Zd. The convergence of T([nx])/n to the time-constant μ(x) defined by μ(x):=\lim_{n\to\infty} T([nx])/n, can be viewed as a problem of homogenization for a discrete Hamilton-Jacobi equation. If the edge weights satisfy 0 < a < τe < b almost surely, μ(x) is a norm on Rd. We borrow techniques from stochastic homogenization to prove a variational formula for the (usual) dual-norm of μ. The variational formula can be used for a variety of purposes; as examples of its use, we solve it exactly for the time-constant when the edge-weights are periodic, and prove bounds in the i.i.d setting. Our results apply to a large class of optimal-control problems on lattices that include directed first-passage percolation and long-range percolation.
Warren Weaver Hall Room 512 at 11:00 am.

Friday, April 26

Arnab Sen, University of Minnesota
Continuous spectra for sparse random graphs.
Abstract:
The limiting spectral distributions of many sparse random graph models are known to contain atoms. But do they also have some continuous part? I will answer this question for several widely studied models of random graphs including Erdos-Renyi random graph G(n, c/n) with c>1, supercritical bond percolation on Z^2 and random graphs with certain degree distributions. I will also persent several open problems. This is joint work with Charles Bordenave and Balint Virag.
Warren Weaver Hall Room 512 at 11:00 am.

Friday, May 3

Amarjit Budhiraja, University of North Carolina at Chapel Hill
Infinity Laplacian and Stochastic Differential Games
Abstract:
A two-player zero-sum stochastic differential game(SDG), motivated by a discrete time random turn game of Peres, Schramm,Sheffield and Wilson(2006) known as the Tug of War, is introduced. The SDG is defined in terms of an m-dimensional state process that is driven by a one-dimensional Brownian motion, played until the state exits the domain. The players controls enter in a diffusion coefficient and in an unbounded drift coefficient of the state process. We show that the game has value, and characterize the value function as the unique viscosity solution of the inhomogeneous infinity Laplace equation introduced in Peres et al. A similar SDG is conjectured for the motion by curvature equation in the plane. Joint work with R. Atar.
Warren Weaver Hall Room 512 at 11:00 am.

Friday, May 10

Combinatorial and probabilistic aspects of dependent percolation.
Abstract:
TBA
Warren Weaver Hall Room 512 at 11:00 am.

## Fall Semester 2012

Friday, September 14

Anirban Basak, Stanford University
Ferromagnetic Ising measures on large locally tree-like graphs.
Abstract:
Consider the ferromagnetic Ising measure on sparse finite graphs converging locally to limiting tree T. In case T is d-regular, it was recently shown by Montanari, Mossel, and Sly that these Ising measures converge locally to symmetric mixture of plus and minus (boundary conditions) Ising measures on T, and for expander graphs, conditioned on positive magnetization these measures converge to plus-boundary condition Ising measure on T. With Amir Dembo, we extend these results, and show universality for a more general, random limiting tree T. In this talk I will review the results of Montanari et al. and discuss the results we obtain.
Warren Weaver Hall Room 512 at 10:00 am.

Saturday & Sunday, September 15-16

Friday, September 21

Louis-Pierre Arguin, Université de Montréal
Poisson-Dirichlet statistics for the extremes of log-correlated Gaussian fields.
Abstract:
Gaussian fields with logarithmically decaying correlations, such as branching Brownian motion and the 2D Gaussian free field, are conjectured to form a new universality class of extreme value statistics (notably in the work of Carpentier & Ledoussal and Fyodorov & Bouchaud). This class is the borderline case between the class of IID random variables (or REM model), and models where correlations start to affect the statistics. In this talk, I will report on the recent rigorous progress in describing the new features of this class. In particular, I will describe the emergence of Poisson-Dirichlet statistics of the Gibbs measure of a non-hierarchical log-correlated Gaussian field similar to the Gaussian free field. This is joint work with Olivier Zindy.
Warren Weaver Hall Room 512 at 10:00 am.

Friday, September 28 - No seminar

Courant Instructor day

Tuesday, October 2 - Note the different Day, Room and Time

Jordan Stoyanov, Newcastle University, UK
Moment Analysis of Probability Distributions.
Abstract:
We deal with the classical moment problem for probability distributions, continuous or discrete, one-dimensional or multidimensional, coming from random variables or stochastic processes. In this talk the emphasis will be on some recent progress in the moment analysis of distributions. Specific topics which will be discussed are:
(a) Comments on the classical Carleman condition and the Krein condition.
(b) New Hardy's criterion for uniqueness in Multidimensional moment problem.
(c) Nonlinear transformations of random data and their moment (in)determinacy.
(d) Moment determinacy of distributions of stochastic processes defined by SDEs.
There will be results, hints for their proof, examples and counterexamples, and also open questions and conjectures.
Warren Weaver Hall Room 1314 at 2:00pm.

Friday, October 5

Shirshendu Chatterjee, Courant Institute
Multiple phase transitions for long-range first-passage percolation on lattices.
Abstract:
Given a graph G with non-negative edge weights, the passage time of a path is the sum of weights of the edges in the path, and the first-passage time to reach u from v is the minimum passage time of a path joining them. We consider a long range first-passage percolation model on ℤd in which, the weight w(x,y) of the edge joining x, y ∈ ℤd has exponential distribution with mean |x-y|α for some fixed α > 0, and the edge weights are independent. We analyze the growth of the set of vertices reachable from the origin within time t, and show that there are four different growth regimes depending on the value of α. Joint work with Partha Dey.
Warren Weaver Hall Room 512 at 10:00am.

Friday, October 12

Antti Knowles, Courant Institute.
Quantum diffusion and delocalization for random band matrices.
Abstract:
I give a summary of recent progress in establishing the diffusion approximation for random band matrices. We obtain a rigorous derivation of the diffusion profile in the regime W > N^{4/5}, where W is the band width and N the dimension of the matrix. As a corollary, we prove complete delocalization of the eigenvectors. Our proof is based on a new self-consistent equation for the Green function. Joint work with L. Erdos, H.T. Yau, and J. Yin.
Warren Weaver Hall Room 512 at 10:00 am.

Wednesday, October 17 - Note the different Day and Time.

Claudio Landim, IMPA, Brazil
Universality of trap models in the ergodic time scale.
Abstract:
Consider a sequence of possibly random graphs $G_N=(V_N, E_N)$, $N\ge 1$, whose vertices's have i.i.d. weights $\{W^N_x : x\in V_N\}$ with a distribution belonging to the basin of attraction of an $\alpha$-stable law, $0<\alpha<1$. Let $X^N_t$, $t \ge 0$, be a continuous time simple random walk on $G_N$ which waits a \emph{mean} $W^N_x$ exponential time at each vertex $x$. Under considerably general hypotheses, we prove that in the ergodic time scale this trap model converges in an appropriate topology to a $K$-process. We apply this result to a class of graphs which includes the hypercube, the $d$-dimensional torus, $d\ge 2$, random $d$-regular graphs and the largest component of super-critical Erdos-Renyi random graphs. Joint work with M. Jara and A. Teixeira
Warren Weaver Hall Room 512 at 11:00 am.

Friday, October 19 - Two consecutive talks.

Raghu Meka, IAS, Princeton
An Invariance Principle for Polytopes
Abstract:
We show an invariance principle (aka limit theorem) for indicator functions of polytopes. Let X be randomly chosen from {-1,1}^n, and let Y be randomly chosen from the standard spherical Gaussian on R^n. For any (possibly unbounded) polytope P formed by the intersection of k halfspaces, we prove that |Pr [X belongs to P] - Pr [Y belongs to P]| < log^{8/5}k * Delta, where Delta is a parameter that is small for polytopes formed by the intersection of "regular" halfspaces (i.e., halfspaces with low influences). The novelty of our invariance principle is the polylogarithmic dependence on k. Previously, only bounds that were at least linear in the number of bounding hyperplanes k were known. Our invariance principle is motivated by problems in computer science and we give two applications of the result to problems in learning theory (bounding the Boolean noise sensitivity of polytopes) and pseudorandom number generation (constructing pseudorandom number generators that fool polytopes). Joint work with Prahladh Harsha and Adam Klivans.
Warren Weaver Hall Room 512 at 10:00 am.

A shape theorem for an epidemic model in dimension larger than 3.
Abstract:
This is a joint work with Enrique Andjel and Nicolas Chabot. We prove a shape theorem for the set of infected individuals in a spatial epidemic model with 3 states (susceptible-infected-recovered) on $Z^d$ for d larger than 3, when there is no extinction of the infection. For this, in order to deal with the travel times of the epidemic, we derive percolation estimates (through dynamic renormalization) for a locally dependent random graph in correspondence with the epidemic model.
Warren Weaver Hall Room 512 at 11:00 am.

Friday, October 26

Philip Protter, Columbia University
Strict Local Martingales and Financial Bubbles.
Abstract:
We will explain how, in the mathematical analysis of financial bubbles, the nuance between a martingale and a local martingale which is not a martingale (called a strict local martingale), plays a fundamental role. This difference allows one to create a statistical test in order to discern, in real time, whether or not a bubble is occurring. The analysis involves strong laws and central limit theorems of the new, Jacod & Barndorff-Nielsen variety, and a new kind of extrapolation theory. The talk is based on joint work with Robert Jarrow and Younes Kchia.
Warren Weaver Hall Room 512 at 10:00 am.

Friday, November 2

No Seminar.

Friday, November 9

Xiuyuan Cheng, Princeton University
The Limiting Spectrum of Random Inner-product Kernel Matrices.
Abstract:
We consider n-by-n matrices whose (i, j)-th entry is f(X_i^T X_j), where X_1, ...,X_n are i.i.d. standard Gaussian random vectors in R^p, and f is a real-valued function. The eigenvalue distribution of these random kernel matrices is studied at the "large p, large n" regime. It is shown that, when both p and n go to infinity, p/n = \gamma which is a constant, and f is properly scaled so that Var(f(X_i^T X_j)) is O(p^{-1}), the spectral density converges weakly to a limiting density on R. While for smooth kernel functions the limiting spectral density has been previously shown to be the Marcenko-Pastur distribution, our analysis is applicable to non-smooth kernel functions, resulting in a new family of limiting densities.
Warren Weaver Hall Room 512 at 10:00 am.

Thursday and Friday, November 15-16

Friday, November 23

Thanksgiving holiday. No Seminar.

Friday, November 30

Alex Drewitz, Columbia University
Effective polynomial ballisticity conditions for random walk in random environment.
Abstract:
The conditions $(T)_\gamma,$ $\gamma \in (0,1),$ which have been introduced by Sznitman in 2002, have had a significant impact on research in random walk in random environment. They require the stretched exponential decay of certain slab exit probabilities for the random walk under the averaged measure and are asymptotic in nature. We show that in all relevant dimensions (i.e., $d \ge 2$), in order to establish the conditions $(T)_\gamma$, it is actually enough to check a corresponding condition $(\mathcal{P})$ of polynomial type on a finite box. In particular, this extends the conjectured equivalence of the conditions $(T)_\gamma,$ $\gamma \in (0,1)$, to all relevant dimensions. Joint work with N. Berger and A.F. Ramírez.
Warren Weaver Hall Room 512 at 10:00 am.

Friday, December 7

David Sivakoff, Duke University
Bootstrap percolation on the Hamming torus.
Abstract:
The Hamming torus of dimension d is the graph with vertices ${1,..., n}^d$ and an edge between any two vertices that differ in a single coordinate. Bootstrap percolation with threshold $\theta$ starts with a random set of open vertices, to which every vertex belongs independently with probability p, and at each time step the open set grows by adjoining every vertex with at least $\theta$ open neighbors. We assume that n is large, d and $\theta$ are fixed, and that p scales as $n^{-\alpha}$ for some $\alpha > 1$, and study the probability that an $i$-dimensional subgraph ever becomes open. For some parameter values we can compute the critical exponent, $\alpha$, exactly, and even compute the spanning probability in the critical window, while for other parameter values we give bounds on the critical exponents.
Warren Weaver Hall Room 512 at 10:00 am.

Friday, December 14

No talk.

## Spring Semester 2012

Friday, February 3

Gabor Kun, Courant Institute
A measurable version of the Lovasz Local Lemma.
Abstract:
The Lovasz Local Lemma (LLL) is one of the basic tools in probabilistic combinatorics. The LLL was only proved for discrete probability spaces. We will prove a measurable version of the LLL. To see what this means consider the following easy corollary of the LLL: Given S_1, ..., S_m r-element subsets of the real numbers, where m<2^r/2e(r+1) the real numbers have a 2-coloring s.t. there is no monochromatic translate of any S_i. The original LLL does not guarantee nice color classes: the Axiom of Choice is used. We will see how to do this in a measurable way. We apply this measurable lemma to the dynamical von Neumann problem highlighting an interesting connection to percolation theory.
Warren Weaver Hall Room 512 at 10:00 am.

Friday, February 10

Tom LaGatta, Courant Institute
Geodesics of Random Riemannian Metrics.
Abstract:
Geodesics are local length-minimizing paths in Riemannian geometry, but it is an interesting question under what conditions they globally minimize length. The Cartan-Hadamard theorem, for example, says that under non-positive curvature assumptions on one's space, geodesics are globally minimizing. In the context of a random metric, one expects a presence of positive curvature, and random geodesics should occasionally run into these positive patches. For perturbations of the Euclidean plane, we have used the point-of-view of the particle technique to show that this is indeed the case, and that a geodesic with randomly selected starting conditions is not minimizing (almost surely). This is joint work with Janek Wehr.
Warren Weaver Hall Room 512 at 10:00 am.

Friday, February 17

Alexey Shashkin, Moscow State University
Limit theorems for geometrical characteristics of Gaussian excursion sets.
Abstract:
Excursion sets of stationary random fields have attracted much attention in recent years. They have been applied to modeling complex geometrical structures in tomography, astrophysics and hydrodynamics. Given a random field and a specified level, it is natural to study geometrical functionals of excursion sets considered in some bounded observation window. Main examples of such functionals are the volume, the surface area and the Euler characteristics. Starting from the classical Rice formula (1945), many results concerning calculation of moments of these geometrical functionals have been proven. There are much less results concerning the asymptotic behavior (as the window size grows to infinity), as random variables considered here depend non-smoothly on the realizations of the random field. In the talk we discuss several recent achievements in this domain, concentrating on asymptotic normality and functional central limit theorems.
Warren Weaver Hall Room 512 at 10:00 am.

Friday, February 24

H. T. Yau, Harvard University
Random Matrix, Beta ensembles, and Dyson Brownian Motion.
Abstract:
The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue statistics of large random matrices in the bulk exhibit universal behavior depending only on the symmetry class of the matrix ensemble. For invariant matrix models, the eigenvalue distributions are given by a log gas with a potential $V$ and inverse temperature $\beta = 1, 2, 4$, corresponding to the orthogonal, unitary and symplectic ensembles. The universality conjecture for invariant ensembles asserts that the local eigenvalue statistics are independent of $V$ for all positive real $\beta$. In this talk, we review the recent progress regarding the universality conjecture for both invariant and non-invariant ensembles. The special role played by the logarithmic Sobolev inequality and Dyson Brownian motion will be discussed.
Warren Weaver Hall Room 512 at 10:00 am.

Friday, March 2

Columbia-Princeton Probability Day.

Confirmed Speakers:
J. C. Mattingly (Duke University)
R. Pemantle (University of Pennsylvania)
L. Saloff-Coste (Cornell University)
M. Damron (Princeton University)

Friday, March 9

Leonid Petrov, Northeastern University
Asymptotics of Uniformly Random Lozenge Tilings of Polygons.
Abstract:
I plan to discuss the model of uniformly random tilings of polygons drawn on the triangular lattice by lozenges of three types. Asymptotic questions about these tilings (when the polygon is fixed and the mesh of the lattice goes to zero) have received a significant attention over the past years. With the help of technique of determinantal point processes, a recent progress has been made for tilings of polygons in a certain class which allows arbitrarily many sides. For these polygons, we establish the conjectural local asymptotics of tilings (leading to ergodic translation invariant Gibbs measures on tilings of the whole plane) and the predicted behavior of interfaces between so-called liquid and frozen phases (leading to the Airy process). Local behavior allows to reconstruct the limit shapes of random stepped surfaces obtained by Cohn, Propp, Kenyon, and Okounkov. As a particular case, these results cover the most investigated case of random boxed plane partitions (when the polygon is a hexagon).
Warren Weaver Hall Room 512 at 10:00 am.

Friday, March 16

No Seminar. Spring Recess.

Friday, March 23

Grigori Olshanski, Institute for Information Transmission Problems, Moscow.
Non-colliding processes with infinitely many particles.
Abstract:
Models of Markov dynamics for N non-colliding particles have been studied in the Random Matrix literature since Dyson's paper (1962) on matrix-valued Brownian motion. However, extension of the theory to the case of infinitely many particles presents substantial difficulties. I will describe a new method of constructing infinite-dimensional Markov dynamics based on some ideas from representation theory of infinite-dimensional groups. This is joint work with Alexei Borodin.
Warren Weaver Hall Room 512 at 10:00 am.

Friday, March 30

Robert Neel, Lehigh University
Minimal surfaces and coupled Brownian motion.
Abstract:
We begin by explaining why stochastic analysis is a natural tool for the study of minimal submanifolds. In this spirit, we then introduce an extrinsic analogue, for minimal surfaces in R^3, of the mirror coupling of two Brownian motions and use it to prove geometric results. The first class of results we look at are strong halfspace-type theorems, in which the goal is to prove that pairs of minimal surfaces, under some conditions, must intersect. Second, we study harmonic functions on minimal surfaces, proving that properly embedded minimal surfaces of bounded curvature admit no non-constant bounded harmonic functions (thus making progress toward a conjecture of Sullivan) and that non-planar minimal graphs are parabolic (thus proving a conjecture of Meeks).
Warren Weaver Hall Room 512 at 10:00 am.

Tuesday, April 3 - Note the different time and place

Dispersion, diffusion (and super-diffusion) of energy in a chain of coupled oscillators.
Abstract:
I will review some results on the dispersion and diffusion of energy in a chain of oscillators whose hamiltonian dynamics is perturbed by stochastic conservative terms. On one dimensional unpinned case the energy super-diffuse. In a weak noise limit, this super-diffusion is described by a self-similar Levy process. But in the hydrodynamic limit we still do not understand the nature of this super-diffusion.
Warren Weaver Hall Room 317 at 11am-12noon.

Friday, April 6

Toby Johnson, Univ. of Washington, Seattle
Growing random regular graphs and the Gaussian Free Field.
Abstract:
The spectral properties of Wigner matrices have been studied intensely. The adjacency matrices of random regular graphs have much in common with Wigner matrices, but they can be different too. For example, the fluctuations of their linear eigenvalue statistics converge to sums of Poissons as the size of the graph tends to infinity, rather than to Gaussians as with Wigner matrices. Alexei Borodin has recently found connections between the eigenvalues of sequences of minors of a Wigner matrix and the Gaussian Free Field. As an analogue to this, we investigate the eigenvalues of a sequence of growing random regular graphs, and we find similar connections. Along the way, we will paint a nice picture of the combinatorial behavior of our growing random regular graphs. This is joint work with Soumik Pal.
Warren Weaver Hall Room 512 at 10:00 am.

Friday, April 13

Edward Waymire, Oregon State University
Dispersion in the Presence of Interfacial Discontinuities.
Abstract:
This talk will focus on probability questions arising in the geophysical and biological sciences concerning dispersion in highly heterogeneous environments, as characterized by abrupt changes (discontinuities) in the diffusion coefficient. Some specific phenomena observed in laboratory and field experiments involving breakthrough curves (first passage times), occupation times, and local times will be addressed within a probabilistic framework largely founded on the Ito-McKean-Feller classic skew Brownian motion and Stroock-Varadhan martingale theory. This is based on joint work with Thilanka Appuhamillage, Vrushali Bokil, Enrique Thomann, and Brian Wood at Oregon State University.
Warren Weaver Hall Room 512 at 10:00 am.

Friday, April 20 - Two consecutive talks.

Sourav Chatterjee, Courant Institute
Invariant measures and the soliton resolution conjecture.
Abstract:
The soliton resolution conjecture for the focusing nonlinear Schrodinger equation (NLS) is the vaguely worded claim that a global solution of the NLS, for generic initial data, will eventually resolve into a radiation component that disperses like a linear solution, plus a localized component that behaves like a soliton or multi-soliton solution. Considered to be one of the fundamental problems in the area of nonlinear dispersive equations, this conjecture has eluded a proof or even a precise formulation till date. I will present a theorem that proves a "statistical version" of this conjecture at mass-subcritical nonlinearity. The proof involves a combination of techniques from large deviations, PDE, harmonic analysis and bare hands probability theory.
Warren Weaver Hall Room 512 at 10:00 am.

Scott Sheffield, MIT
Imaginary Geometry and SLE.
Abstract:
It turns out that there is a pretty natural way to say what a two-dimensional surface with “imaginary Gaussian curvature” should be. There is also a pretty natural way to say what a “random” imaginary surface should be (closely related to the “real” random surfaces of Liouville quantum gravity). I will give a high-level overview of these topics, illustrated by computer simulations. I will then explain how one can use this perspective to resolve several open problems about a famous family of random fractals — the so-called SLE curves. The talk is based on recent joint work with Jason Miller.
Warren Weaver Hall Room 512 at 11:00 am.

Friday, April 27 - Two consecutive talks.

Alexei Borodin, MIT
Directed random polymers and Macdonald processes.
Abstract:
The goal of the talk is to survey recent progress in understanding statistics of certain exactly solvable growth models, particle systems, directed polymers in one space dimension, and stochastic PDEs. A remarkable connection to representation theory and integrable systems is at the heart of Macdonald processes, which provide an overarching theory for this solvability. This is based on joint work with Ivan Corwin.
Warren Weaver Hall Room 512 at 10:00 am.

Ilya Goldsheid, Queen Mary, University of London
Quenched sub-diffusive 1D random walks in random environment.
Abstract:
In the sub-diffusive regime, the 1D random walks in random environment do not have quenched distributional limits (Peterson & Zeitouni, Ann. Prob. 2009). Nevertheless, the limiting behaviour of such random walks can be described as a random linear combination of standard exponential random variables. The corresponding coefficients form a point Poisson process defined on the space of environments and having an explicit density. As a corollary, one obtains a new proof of the classical annealed theorem of Kesten-Kozlov-Spitzer as well as the just mentioned result of Peterson-Zeitouni. This is joint work with D. Dolgopyat.
Warren Weaver Hall Room 512 at 11:00 am.

Friday, May 4

Lingjiong Zhu, Courant Institute
Limit Theorems for Nonlinear Hawkes Processes.
Abstract:
Hawkes process is a self-exciting point process with clustering effect whose intensity depends on its entire past history. It has wide applications in neuroscience, fi nance and many other fi elds. Linear Hawkes process has an immigration-birth representation and can be computed more or less explicitly. It has been extensively studied in the past and the limit theorems are well understood. On the contrary, nonlinear Hawkes process lacks the immigration-birth representation and is much harder to analyze. In this talk, we will discuss a functional central limit theorem and large deviations for nonlinear Hawkes process.
Warren Weaver Hall Room 512 at 10:00 am.

Friday, May 11

Dmytro Karabash, Courant Institute
On Properties of Hawkes Process.
Abstract:
This talk is on a particular type of self-exciting process. In focus of this talk is study of stability under coefficients previously not touched in literature while local tails are proved as lemma. The tree structure and domination structure are observed and explicitly used in proofs. The main stability result lifts condition of 1-Lipschitz continuity that was previously imposed in Brémaud-Massoulié. First result replaces 1-Lipschitz condition with continuous modulus of continuity and second result allows jumps under some additional but natural assumptions. Generalizations and ramifications are provided.
Warren Weaver Hall Room 512 at 10:00 am.

Friday, May 18

Mikko Stenlund, University of Rome "Tor Vergata"
An invariance principle for Sinai billiards with random scatterers.
Abstract:
Understanding the statistical properties of the aperiodic planar Lorentz gas stands as a grand challenge in the theory of dynamical systems. We study a greatly simplified but related model, popularized by Joel Lebowitz, in which a scatterer configuration on the torus is randomly updated between collisions. Taking advantage of recent progress in the theory of time-dependent billiards on the one hand and in probability theory on the other, we prove a vector-valued almost sure invariance principle for the model. Notably, the configuration sequence can be weakly dependent and non-stationary. We also obtain a new invariance principle for Sinai billiards (the case of fixed scatterers) with time-dependent observables, and improve the accuracy and generality of existing results. The article is available at http://arxiv.org/abs/1210.0902
Warren Weaver Hall Room 512 at 10:00 am.

## Fall Semester 2011

Minerva Foundation Lectures at Columbia University, September 7--15

Denis Talay, INRIA Sophia Antipolis
Model Risk: Modeling, Analysis, Control and Numerics.
Abstract:
The objective of these lessons is to show that model risk analysis, particularly financial model risk analysis, opens new interesting stochastic analysis problems, to present recent mathematical and numerical techniques to tackle them, and to analyze mathematically some robust strategies which, issued from the technical analysis, do not rely on a specific mathematical model. We will also present a selection of challenging open questions. Various theories will be used, such as statistics of random processes, stochastic control, Malliavin calculus, backward stochastic differential equations, viscosity solutions of nonlinear Partial Differential equations. However the course will be self-contained and, whenever possible, the proofs will be fully detailed.

Friday, September 9

Denis Talay, INRIA Sophia Antipolis
Stochastic Approaches for Parabolic and Elliptic Diffraction Equations.
Abstract:
We consider partial differential equations of parabolic or elliptic type involving a divergence form operator with a discontinuous coefficient and a compatibility transmission condition. We prove existence and uniqueness results by stochastic methods which also allow us to develop or justify low complexity Monte Carlo numerical resolution methods and to get sharp convergence rate estimates.
Warren Weaver Hall Room 317 at 10:00 am.

Friday, September 16 - Two consecutive talks.

Helmut Katzgraber, Texas A&M University and ETH Zurich
Universality in Levy spin glasses.
Abstract:
Spin glasses are paradigmatic models that deliver concepts relevant for a variety of systems. Concepts from the solution of the mean-field model, such as ergodicity breaking, aging and ultrametricity have been applied to realistic short-range spin-glass models as well as to fields as diverse as structural biology, geology, computer science and even financial analysis. However, despite ongoing research spanning several decades in the area of glassy systems, there remain many fundamental open questions. Rigorous analytical results are difficult to obtain for spin-glass models, in particular for realistic short-range systems. Therefore large-scale numerical simulations are the tool of choice. After presenting a brief overview of spin glasses, the concept of universality, a cornerstone of statistical physics, is discussed. Although it is well established numerically that universality is not violated for nearest-neighbor spin glasses with compact disorder distributions (e.g., Gaussian and bimodal), some studies suggest that this might not be the case when the disorder distributions are broad, as in the case of the Levy distribution. Using large-scale Monte Carlo simulations that combine parallel tempering with specialized cluster moves, as well as innovative scaling techniques, we show that Levy spin glasses do obey universality.
Work done in collaboration with J. C. Andresen and K. Janzen.
Warren Weaver Hall Room 317 at 10:00 am.

Amir Dembo, Stanford University
Lecture series on "Gibbs measures and phase transitions on sparse random graphs"
Abstract:
Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years, considerable progress has been achieved by viewing these distributions as Gibbs measures and applying to their study heuristic tools from statistical physics. We will review this approach and provide some results towards a rigorous treatment of these problems.
Outline for Day 1 of 4:
Models on graphs, phase transitions, gibbs measures, mean field equations and approximation by trees. Reference: Chapter 1 & 2 of Dembo and Montanari, "Gibbs measures and phase transitions on sparse random graphs", Brazilian J. of Probab. and Stat. 24 (2010), pp. 137-211.
Warren Weaver Hall Room 317 at 11:00 am.

Friday, September 23 - Two consecutive talks.

Dmitry Ioffe, Technion
Critical drifts for random walks in attractive potentials.
Abstract:
Self-attractive random walks (polymers) undergo a phase transition in terms of the applied drift: If the drift is strong enough, then the walk is ballistic, whereas in the case of small drifts self-attraction wins and the walk is sub-ballistic. We show that, in any dimension larger than one, this transition is of first order. In fact, we prove that the walk is already ballistic at critical drifts, and establish the corresponding LLN and CLT. Joint work with Yvan Velenik.
Warren Weaver Hall Room 317 at 10:00 am.

Amir Dembo, Stanford University
Lecture series on "Gibbs measures and phase transitions on sparse random graphs"
Abstract:
Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years, considerable progress has been achieved by viewing these distributions as Gibbs measures and applying to their study heuristic tools from statistical physics. We will review this approach and provide some results towards a rigorous treatment of these problems.
Outline for Day 2 of 4:
Ferromagnetic Ising model on sparse graphs: Convergence to the tree measure, limiting free energy, belief propagation algorithm and phase coexistence. Reference: Chapter 2 of Dembo and Montanari, "Gibbs measures and phase transitions on sparse random graphs", Brazilian J. of Probab. and Stat. 24 (2010), pp. 137-211.
Warren Weaver Hall Room 317 at 11:00 am.

Friday, September 30 - Two consecutive talks.

Mykhaylo Shkolnikov, Stanford University
On diffusions interacting through their ranks.
Abstract:
We will discuss systems of diffusion processes on the real line, in which the dynamics of every single process is determined by its rank in the entire particle system. Such systems arise in mathematical finance and statistical physics, and are related to heavy-traffic approximations of queueing networks. Motivated by the applications, we address questions about invariant distributions, convergence to equilibrium and concentration of measure for certain statistics, as well as hydrodynamic limits and large deviations for these particle systems. Parts of the talk are joint work with Amir Dembo, Tomoyuki Ichiba, Soumik Pal and Ofer Zeitouni.
Warren Weaver Hall Room 317 at 10:00 am.

Amir Dembo, Stanford University
Lecture series on "Gibbs measures and phase transitions on sparse random graphs"
Abstract:
Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years, considerable progress has been achieved by viewing these distributions as Gibbs measures and applying to their study heuristic tools from statistical physics. We will review this approach and provide some results towards a rigorous treatment of these problems.
Outline for Day 3 of 4:
Finite-size scaling, the ODE method and its refinement through diffusion limit and strong approximation. XORSAT - an application to coding theory. Reference: Chapter 6 of Dembo and Montanari, "Gibbs measures and phase transitions on sparse random graphs", Brazilian J. of Probab. and Stat. 24 (2010), pp. 137-211.
Warren Weaver Hall Room 317 at 11:00 am.

Friday, October 7 - Two consecutive talks.

Olivier Bernardi, MIT
Computing the moments of the GOE bijectively.
Abstract:
The GOE, or Gaussian Orthogonal Ensemble, is a Gaussian measure on the set of orthogonal matrices. We consider the problem of finding the nth moment of the eigenvalues of the matrices in the GOE. It turns out that this problem is closely related to a question about the different ways of gluing the edges of a 2n-gon in pairs so as to create a surface without boundary. More precisely, among the (2n)!/n! possible gluings, how many times does one get each surface (considered up to homeomorphism)?
In this talk, we will recall the connection between the two questions, and present a bijective solution. Our results are analogous to the one obtained by Harer and Zagier (1986) about the gluings of a 2n-gon giving an orientable surface (or in matrix terms, about the Gaussian Unitary Ensemble). We also recover a recurence formula for the moments of the GOE recently obtained by Ledoux.
Warren Weaver Hall Room 317 at 10:00 am.

Amir Dembo, Stanford University
Lecture series on "Gibbs measures and phase transitions on sparse random graphs"
Abstract:
Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years, considerable progress has been achieved by viewing these distributions as Gibbs measures and applying to their study heuristic tools from statistical physics. We will review this approach and provide some results towards a rigorous treatment of these problems.
Outline for Day 4 of 4:
Reconstruction on trees and random graphs. Constraint satisfaction problems, Clustering phase transition . Reference: Chapter 5 of Dembo and Montanari, "Gibbs measures and phase transitions on sparse random graphs", Brazilian J. of Probab. and Stat. 24 (2010), pp. 137-211.
Warren Weaver Hall Room 317 at 11:00 am.

Friday, October 14 - Two consecutive talks.

From random interlacements to coordinate and infinite cylinder percolation.
Abstract:
During the talk I will focus on the connectivity properties of three models with long (infinite) range dependencies: Random Interlacements, percolation of the vacant set in infinite rod model and Coordinate percolation. The latter model have polynomial decay in sub-critical and super-critical regime in dimension 3. I will explain the nature of this phenomenon and why it is difficult to handle these models technically. In the second half of the talk I will present key ideas of the multi-scale analysis which allows to reach some conclusions. At the end I will discuss applications and several open problems.
Warren Weaver Hall Room 317 at 10:00 am.

Rigidity of 3-colorings of the d-dimensional discrete torus.
Abstract:
We prove that a uniformly chosen proper coloring of Z_{2n}^d with 3 colors has a very rigid structure when the dimension d is sufficiently high. The coloring takes one color on almost all of either the even or the odd sub-lattice. In particular, one color appears on nearly half of the lattice sites. This model is the zero temperature case of the 3-states anti-ferromagnetic Potts model, which has been studied extensively in statistical mechanics. The proof involves results about graph homomorphisms and various combinatorial methods, and follows a topological intuition. Joint work with Ron Peled.
Warren Weaver Hall Room 317 at 11:05 am.

Friday, October 21 - Two consecutive talks.

Amir Dembo, Stanford University
Factor models on locally tree-like graphs
Abstract:
Consider factor (graphical) models on sparse graph sequences that converge locally to a random tree T. Using a novel interpolation scheme we prove existence of limiting free energy density under uniqueness of relevant Gibbs measures for the factor model on T. We demonstrate this for Potts and independent sets models and further characterize this limit via large-deviations type minimization problem and provide an explicit formula for its solution, as the Bethe free energy for a suitable fixed point of the belief propagation recursions on T (thereby rigorously generalize heuristic calculations by statistical physicists using replica'' or cavity'' methods). This talk is based on a joint work with Andrea Montanari and Nike Sun.
Warren Weaver Hall Room 317 at 10:00 am.

Ivan Corwin, Microsoft Research - New England, MA
Brownian Gibbs line ensembles.
Abstract:
The Airy line ensemble arises in scaling limits of growth models, directed polymers, random matrix theory, tiling problems and non-intersecting line ensembles. This talk will mainly focus on the "non-intersecting Brownian Gibbs property" for this infinite ensemble of lines. Roughly speaking, the measure on lines is invariant under resampling a given curve on an interval according to a Brownian Bridge conditioned to not intersect the above of below labeled curves. This property leads to the proof of a number of previously conjectured results about the top line of this ensemble. We will also briefly touch on the KPZ line ensemble, which arises as the scaling limit of a diffusion defined by the Doob-h transform of the quantum Toda lattice Hamiltonian. The top labeled curve of this KPZ ensemble is the fixed time solution to the famous Kardar-Parisi-Zhang stochastic PDE. This line ensemble has a "softer" Brownian Gibbs property in which resampled Brownian Bridges may cross the lines above and below, but at exponential energetic cost.
Warren Weaver Hall Room 317 at 11:05 am.

Friday, October 28 - Two consecutive talks.

Giambattista Giacomin, Université Paris Diderot
Coherence stability and effect of random natural frequencies in populations of coupled oscillators.
Abstract:
We consider the (noisy) Kuramoto model, that is a population of N oscillators, or rotators, with mean-field interaction. Each oscillator has its own natural frequency, which is chosen randomly (quenched disorder) and it is stirred by Brownian motion. In the limit of large N this model is accurately described by a (deterministic) Fokker-Panck equation. We study this equation and obtain quantitatively sharp results in the limit of weak disorder. We show that, in general, the oscillators synchronize (for sufficiently strong interaction) around a common rotating phase, whose frequency is sharply estimated. These results are obtained by identifying the stable hyperbolic manifold of stationary solutions of an associated non disordered model and by exploiting the robustness of hyperbolic structures under suitable perturbations. The method applies also to cases in which the single rotator dynamics is not just a (random) rotation: in fact, to a certain extent, the single rotator dynamics can be arbitrary.
Warren Weaver Hall Room 317 at 10:00 am.

Edward Waymire, Oregon State University
Tree polymers under weak/strong disorder.
Abstract:
Tree polymers are simplifications of 1+1 dimensional lattice polymers made up of polygonal paths of a (nonrecombining) binary tree having random path probabilities. As in the case of lattice polymers, the path probabilities are (normalized) products of i.i.d. positive weights. The a.s. probability laws of these paths are of interest under weak and strong types of disorder. Some recent results, speculation and conjectures will be presented for this class of models under both weak and strong disorder conditions. In particular results are included that suggest an explicit formula for the asymptotic variance of the free end'' under strong disorder. This is based on joint work with Stanley Williams and Torrey Johnson.
Warren Weaver Hall Room 317 at 11:05 am.

Tuesday, November 1 - Special talk

Lorenzo Bertini, Universita' di Roma La Sapienza
Large deviation principle of the empirical current for Markov processes.
Abstract:
We consider a continuous time Markov chain on a countable state space and extend the classical Donsker-Varadhan large deviation principle for the empirical measure by considering also the empirical flow. We then discuss the application to the Gallavotti-Cohen functional, whose associated large deviation principle can be obtained by projection. We finally illustrate briefly the analogous results for diffusion processes on R^n.
Warren Weaver Hall Room 1314 at 2:00 pm.

Friday, November 4 - Two consecutive talks.

Roberto Cyril, Université de Marne-La-Vallée
Some rigorous result on the East Model.
Abstract:
We will consider a special example of one dimensional kinetically constrained model, the East model. We will start by briefly reviewing some of the known results on the dynamics : spectral gap, persistence function, long-time behavior starting from non-equilibrium. Then, we will focus on the low temperature non-equilibrium dynamics which follows a quench from an initial distribution which is different from the reversible one. This setting has been extensively studied in physics literature: on the basis of heuristic arguments and numerical simulations it was observed that dynamics can be approximated by an irreversible coarsening process for the domains (intervals separating consecutive vacancies) with a peculiar hierarchical structure. We will explain how, provided the initial distribution of the domains is a renewal process, this approximation can be made rigorous and how, by analyzing the asymptotic behavior of the coalescence process, one can prove a staircase behavior for the persistence function, an aging behavior for the correlation function and give a sharp description on the statistics of the intervals separating consecutive vacancies. (based on a series of papers in collaboration with N. Cancrini, A. Faggionato, F. Martinelli and C. Toninelli).
Warren Weaver Hall Room 317 at 10:00 am.

Nike Sun, Stanford University
Potts and independent set models on d-regular graphs.
Abstract:
We consider the ferromagnetic Potts on typical d-regular graphs, and the independent set model on typical bipartite d-regular graphs, with graph size tending to infinity. We show that the replica symmetric (Bethe) prediction applies for all parameter values in these two models. In this talk I will describe some of the proof techniques, which will give an indication of the contrast with the anti-ferromagnetic Potts model and the independent set model at high fugacity on non-bipartite graphs, where the Bethe prediction is known to fail.This is joint work with Amir Dembo, Andrea Montanari, and Allan Sly.
Warren Weaver Hall Room 317 at 11:05 am.

Wednesday, November 9 - Special talk

Alan Sokal, New York University
Some wonderful conjectures at the boundary between analysis, combinatorics and probability.
Abstract:
I discuss some analytic and combinatorial properties (most of which are at present only conjectural) of the entire function
F(x,y) = Σn≥0 xn/n!   yn(n-1)/2 .
This function (or formal power series) arises in numerous problems in enumerative combinatorics, notably in the enumeration of connected graphs, and in statistical mechanics in connection with the Potts model on the complete graph (mean-field'' or Curie--Weiss Potts model). This circle of problems also touches on the theory of integrable systems in classical mechanics (Calogero--Moser system). If time permits I will discuss an analogous problem for the "partial theta function":
Θ0(x,y) = Σn≥0 xn yn(n-1)/2
in this case some striking results can be proven, by using identities for q-series. For details, see http://www.maths.qmul.ac.uk/~pjc/csgnotes/sokal/ and http://arxiv.org/abs/1106.1003.
Warren Weaver Hall Room 517 at 10:00 am.

Friday, November 11 - Two consecutive talks.

Subhrosekhar Ghosh, UC Berkeley
What does a Point Process Outside a Domain tell us about What's Inside?
Abstract:
In a Poisson point process we have independence between disjoint spatial domains, so the points outside a disk give us no information on the points inside. The story gets a lot more interesting for processes with stronger spatial correlation. In the case of Ginibre ensemble, a process arising from eigenvalues of random matrices, we prove that the outside points determine exactly the number of points inside, and further, we demonstrate that they determine nothing more. In the case of zero ensembles of Gaussian power series, we prove that the outside points determine exactly the number and the centre of mass of the inside points, and nothing further. These phenomena suggest a certain hierarchy of point processes based on their rigidity; Poisson, Ginibre and the Gaussian power series fit in at levels 0, 1 and 2 in this ladder.
Time permitting, we will also look at some interesting consequences of our results, with applications to continuum percolation, reconstruction of Gaussian entire functions, and others. This is based on joint work with Fedor Nazarov, Yuval Peres and Mikhail Sodin.
Warren Weaver Hall Room 317 at 10:00 am.

Barry McCoy, State University of New York, Stony Brook
The Romance of the Ising model
Abstract:
The essence of romance is mystery. In this talk I will explore the meaning of this for the Ising model, beginning in 1946 with Bruria Kaufman and Willis Lamb, to the wedding of the Ising model with Painlevé functions, to the discovery of a possible natural boundary in the susceptibility and concluding with recent work (and mysteries) on the factorization of the form factor expansion and the relation of the diagonal susceptibility to p+1Fp hypergeometric functions, modular forms and particular Calabi-Yau equations.
Warren Weaver Hall Room 317 at 11:05 am.

Thursday and Friday, November 17-18

Invited speakers:

Vlada Limic, Université de Provence, Some progress in understanding the small-time behavior of exchangeable coalescents
Eyal Lubetzky, Microsoft Research, From entropic repulsion to the shape of (2+1)-dimensional SOS
Gregory Miermont, Université Paris Sud and University of British Columbia, Random maps and their scaling limits
Jeremy Quastel, University of Toronto, Exact solutions in random growth and directed polymers

Held at Courant Institute of Mathematical Sciences, NYU. More details at the seminar website.

Friday, November 25

Thanksgiving holiday. No Seminar.

Friday, December 2 - Two consecutive talks.

Leonid Koralov, University of Maryland
Polymer measures and branching diffusions.
Abstract:
We study two problems related by a common set of techniques. In the first problem, we consider a model for the distribution of a long homopolymer in a potential field. For various values of the temperature, including those at or near the critical value, we consider the limiting behavior of the polymer when its size tends to infinity. In the second problem, we investigate the long-time evolution of branching diffusion processes in inhomogeneous media. The qualitative behavior of the processes depends on the intensity of the branching. In the super-critical case, we describe the asymptotics of the number of particles in a given domain and describe the growth of the region containing the particles. In the sub-critical regime, we describe the limiting distribution of the total number of particles.
Warren Weaver Hall Room 317 at 10:00 am.

Michael Damron, Princeton University
A simplified proof of the relation between scaling exponents in first-passage percolation.
Abstract:
In first passage percolation, we place i.i.d. non-negative weights on the nearest-neighbor edges of Z^d and study the induced random metric. A long-standing conjecture gives a relation between two "scaling exponents": one describes the variance of the distance between two points and the other describes the transversal fluctuations of optimizing paths between the same points. This is sometimes referred to as the "KPZ relation." In a recent breakthrough work, Sourav Chatterjee proved this conjecture using a strong definition of the exponents. I will discuss work I just completed with Tuca Auffinger, in which we introduce a new and intuitive idea that replaces Chatterjee's main argument and gives an alternative proof of the relation. One advantage of our argument is that it does not require a certain non-trivial technical assumption of Chatterjee on the weight distribution.
Warren Weaver Hall Room 317 at 11:05 am.

Friday, December 9 - Two consecutive talks.

Phase transitions in complex networks.
Abstract:
We consider the competition between structures in large simple graphs, for instance the competition between the density of edges and the density of triangles. Using "graph limits" to control the asymptotics of probability distributions on graphs, one finds well defined phases in the parameter space, with perfectly sharp transitions, in close analogy with the liquid/gas and fluid/solid transitions of statistical mechanics.
Warren Weaver Hall Room 317 at 10:00 am.

Yuri Kifer, Hebrew University
A Zoo of Nonconventional Limit Problems.
Abstract:
We discuss various limit theorems for "nonconventional" sums of the form Σ1≤n≤N B(ξ(q1(n)), ξ(q2(n)), ..., ξ(q(n))) where ξ(n), n ≥ 0 is either a Markov chain or a hyperbolic (expanding, subshift of finite type etc.) transformation (i.e. then ξ(n) = Tnx) while qi(n), i ≤ k are linear and qj(n), k < j ≤ ℓ grow faster than linearly. The motivation for this study comes, in particular, from many papers about nonconventional ergodic theorems appeared in the last 30 years. Among our results are central limit theorem, large deviations, averaging and Poisson type limit theorems. We will talk also about some "nonconventional" multifractal formalism type problems computing the Hausdorff dimension of sets of numbers whose expansions have prescribed frequencies of combinations of digits in places qj(n), j = 1, 2, ..., ℓ; n ≥ 1.
Warren Weaver Hall Room 317 at 11:05 am.

## Spring Semester 2011

Friday, January 28

Pieter Trapman, Stockholm University
Long-range percolation on the hierarchical lattice.
Abstract:
The hierarchical lattice of order N, may be seen as the leaves of an infinite regular N-tree, in which the distance between two vertices is the distance to their most recent common ancestor in the tree. We create a random graph by independent long-range percolation on the hierarchical lattice of order N: The probability that a pair of vertices at (hierarchical) distance R share an edge depends only on R and is exponentially decaying in R, furthermore the presence or absence of different edges are independent. We give criteria for percolation (the presence of an infinite cluster) and we show that in the supercritical parameter domain, the infinite component is unique. Furthermore, we show the percolation probability (the density of the infinite cluster) is continuous in the model parameters. In particular, there is no percolation at criticality. Joint work with Slavik Koval and Ronald Meester
Warren Weaver Hall Room 317 at 10:00 am.

Friday, February 4

S.R.S. Varadhan, Courant Institute, New York University
Large deviations for dense random graphs.
Abstract:
In this joint work with Sourav Chatterjee we investigate the large deviation properties of various subgraph counts in random graphs $G(n,p)$ having $n$ vertices with every unoriented edge having independently probability $p$ of being present. The large deviation is carried out in the space of "graph limits" with "cut topology" that allows for continuous contraction to subgraph counts. For example, questions like what is the most likely way the triangle count can be higher (or lower) by a factor from their expected values are answered and exhibit some qualitative changes in behavior as the parameters vary. Finally, there is a curious application to random matrices.
Warren Weaver Hall Room 317 at 10:00 am.

Friday, February 11

Ioannis Karatzas, Columbia University
Stable diffusions interacting through their ranks, as models for large equity markets.
Abstract:
We introduce and study ergodic multidimensional diffusion processes interacting through their ranks. These interactions give rise to invariant measures which are in broad agreement with stability properties observed in large equity markets over long time-periods. The models we develop assign growth rates and variances that depend on both the name (identity) and the rank (according to capitalization) of each individual asset. Such models are able realistically to capture critical features of the observed stability of capital distribution over the past century, all the while being simple enough to allow for rather detailed analytical study. The methodologies used in this study touch upon the question of triple points for systems of interacting diffusions; in particular, some choices of parameters may permit triple (or higher-order) collisions to occur. We show, however, that such multiple collisions have no effect on any of the stability properties of the resulting system. This is accomplished through a detailed analysis of collision local times. The theory we develop has connections with the analysis of Queueing Networks in heavy traffic, as well as with models of competing particle systems in Statistical Mechanics such as the Sherrington-Kirkpatrick model for spin-glasses.
Warren Weaver Hall Room 317 at 10:00 am.

Friday, February 18

Jason Miller, Microsoft Research
CLE(4) and the Gaussian Free Field.
Abstract:
The discrete Gaussian free field (DGFF) is the Gaussian measure on real-valued functions h(.) on a bounded subset D of the two dimensional integer lattice, whose covariance is given by the Green's function for simple random walk. The graph of h(.) is a random surface which serves as a physical model for an effective interface. We show that the collection of random loops given by the level sets of the DGFF at any height converges in the fine-mesh scaling limit to a family of loops which is invariant under conformal transformations when D is a lattice approximation of a non-trivial simply connected domain. In particular, there exists λ>0 such that the level sets whose height is an odd integer multiple of lambda converges to a nested conformal loop ensemble with parameter κ=4 (so-called CLE(4)), a conformally invariant measure on loops which locally look like SLE(4). Using this result, we give a coupling of the continuum Gaussian free field (GFF), the fine-mesh scaling limit of the DGFF, and CLE(4) such that the GFF can be realized as a functional of CLE(4) and conversely CLE(4) can be made sense as a functional of the GFF. Based on joint work with Scott Sheffield.
Warren Weaver Hall Room 317 at 10:00 am.

Friday, February 25

Will Perkins, Courant Institute, New York University
The Bohman-Frieze Process.
Abstract:
The Bohman-Frieze process is a simple modification of the Erdős-Rényi random graph that adds dependence between the edges biased in favor of joining isolated vertices. We present new results on the phase transition of the Bohman-Frieze process and show that qualitatively it belong to the same class as the Erdős-Rényi process. The results include the size and structure of small components in the barely sub- and supercritical time periods. We will also mention a class of random graph processes that seems to exhibit markedly different critical behavior.
Warren Weaver Hall Room 317 at 10:00 am.

Friday, March 4 - Two consecutive talks.

Jian Ding, UC Berkeley
Cover times, blanket times, and the Gaussian free field.
Abstract:
The cover time of a finite graph (the expected time for the simple random walk to visit all the vertices) has been extensively studied, yet a number of fundamental questions concerning cover times have remained open. Aldous and Fill (1994) asked whether there is a deterministic polynomial-time algorithm that computes the cover time up to an O(1) factor. Winkler and Zuckerman (1996) defined the blanket time (when the empirical distribution is within a factor of 2, say, of the stationary distribution) and conjectured that the blanket time is always within O(1) of the cover time. The best approximation factor found so far for both these problems was (log log n)^2 for n-vertex graphs, due to Kahn, Kim, Lovasz, and Vu (2000). We show that the cover time of a graph, appropriately normalized, is proportional to the expected maximum of the (discrete) Gaussian free field on G. We use this connection and Talagrand's majorizing measures theory to deduce a positive answer to the question of Aldous and Fill and to establish the conjecture of Winkler and Zuckerman. These results extend to arbitrary reversible finite Markov chains. No prior knowledge of Talagrand's theory or of cover times will be assumed. This is joint work with James Lee and Yuval Peres.
Warren Weaver Hall Room 317 at 10:00 am.

Ivan Corwin, Courant Institute
Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions.
Abstract:
We consider the solution of the stochastic heat equation with multiplicative noise and delta function initial condition whose logarithm, with appropriate normalizations, is the free energy of the continuum directed polymer, or the solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions. We prove explicit formulas for the one-dimensional marginal distributions -- the crossover distributions -- which interpolate between a standard Gaussian distribution (small time) and the GUE Tracy-Widom distribution (large time). The proof is via a rigorous steepest descent analysis of the Tracy-Widom formula for the asymmetric simple exclusion with anti-shock initial data, which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit. The limit also describes the crossover behaviour between the symmetric and asymmetric exclusion processes.
Warren Weaver Hall Room 317 at 11:15 am.

Friday, March 11

Asaf Nachmias , MIT
The phase transition in percolation on the Hamming cube.
Abstract:
Consider percolation on the Hamming cube {0,1}^n at the critical probability p_c (at which the expected cluster size is 2^{n/3}). It is known that if p=p_c(1+O(2^{-n/3}), then the largest component is of size roughly 2^{2n/3} with high probability and that this quantity is non-concentrated. We show that for any sequence eps(n) such that eps(n)>>2^{-n/3} and eps(n)=o(1) percolation at p_c(1+eps(n)) yields a largest cluster of size (2+o(1))eps(n)2^n. This result settles a conjecture of Borgs, Chayes, van der Hofstad, Slade and Spencer. Joint work with Remco van der Hofstad.
Warren Weaver Hall Room 317 at 10:00 am.

Friday, March 18

Spring recess. No Seminar.

Friday, March 25

Tonći Antunović, UC Berkeley
Some path properties of Brownian motion with variable drift.
Abstract:
If B is a Brownian motion and f is a function in the Dirichlet space, then by Cameron-Martin theorem, the process (B - f) has the same almost sure path properties as B. In this talk we will present some properties of the image and the zero set of Brownian motion perturbed by certain less regular drifts f (examples include Hilbert curves and Cantor functions). Based on joint works with Krzysztof Burdzy, Yuval Peres, Julia Ruscher and Brigitta Vermesi.
Warren Weaver Hall Room 317 at 10:00 am.

Friday, April 1

Antonio Auffinger, Courant Institute
Heavy Tailed Random Matrices and Directed Polymers.
Abstract:
The sum of iid random variables properly scaled does not always converge to a Gaussian distribution as in the CLT. If they are heavy tailed the scaling constant changes and the limit law is no longer Gaussian. In this talk I will show analogous results in three different models where different and new limiting processes arise: the largest eigenvalue of random matrices, the last passage time in last passage percolation and the path measure in Directed Polymers in Random Environments. The main goal is to compare the domain of attraction of (conjectured/proved) universality phenomena of these models. This talk is based on joint works with G. Ben Arous (Courant) and O. Louidor (UCLA) and S.Péché (Grenoble).
Warren Weaver Hall Room 317 at 10:00 am.

Friday, April 8

Allan Sly, Microsoft Research
Phase transitions and the complexity of counting.
Abstract:
Phase transitions have been conjectured to determine the computational complexity of a number of natural combinatorial counting problems. In this talk I will discuss the discrete hardcore model and its relationship to counting the independent sets of a graph. We show that unless NP=RP there is no polynomial time approximation scheme for the partition function of the hardcore model (a weighted sum of independent sets) on graphs of maximum degree d for fugacity \lambda_c<\lambda<\lambda_c + \epsilon(d) where \lambda_c is the uniqueness threshold on the d-regular tree. Weitz produced an efficient algorithm for approximating the partition function when 0 < \lambda < \lambda_c(d) so this result demonstrates that the computational threshold exactly coincides with the statistical physics phase transition thus confirming a conjecture of Mossel, Weitz and Wormald. The proof hinges on a detailed understanding of the distribution of the hardcore model on random bi-partite graphs using the small graph conditioning theorem from combinatorics and point to set correlations of extremal Gibbs measures.
Warren Weaver Hall Room 317 at 10:00 am.

Courant Lecture by Persi Diaconis at 11:30 am in 109 WWH.

Friday, April 15

Columbia-Princeton Probability Day

Schedule (tentative)
 09:00-10:00 AM Registration and continental breakfast 10:00-11:00 AM Alexei Borodin 11:00-12:00 AM Shige Peng 12:00-01:30 PM Lunch 01:30-02:30 PM Yakov Sinai 02:30-03:30 PM Horng-Tzer Yau 03:30-04:00 PM Coffee break 04:00-04:25 PM Antonio Auffinger 04:25-04:50 PM Ilya Vinogradov 04:50-05:15 PM Hana Kogan
Robertson Hall, Room 001 on the Princeton University campus. Full program and map here.

Friday, April 22

Patricia Gonçalves, University of Minho.
Scaling limits of additive functionals of exclusion processes.
Abstract:
In this talk I will consider exclusion processes denoted by (ηt)t≥0, evolving on ℤ and starting from the invariant state: the Bernoulli product measure (νρ)ρ∈[0,1]. The goal of the talk consists in establishing scaling limits of the functional
Γt(f)   := ∫[0,t]f(ηs) ds

for proper local functions f. When f(η) := η(x), the functional Γt(f) is called the occupation time of the origin. I will present a method that was recently introduced in Goncalves and Jara (10') "Universality of the KPZ equation", from which we derive a local Boltzmann-Gibbs Principle for a class of exclusion processes. For the occupation time of the origin, this principle says that the functional Γt(f) is very well approximated to the density of particles. As a consequence, the scaling limits of Γt(f) follow from the scaling limits of the density of particles. As examples I will present the symmetric simple exclusion, the mean-zero exclusion and the weakly asymmetric simple exclusion. For the latter, when the asymmetry is strong enough such that the fluctuations of the density of particles are given by the KPZ equation, we establish the limit of Γt(f) in terms of this solution. The case of asymmetric simple exclusion will also be discussed. This is a joint work with Milton Jara (IMPA-Brazil).
Warren Weaver Hall Room 317 at 10:00 am.

Friday, April 29

Jiří Černý, ETH, Zürich.
Vacant set of random walk on (random) graphs.
Abstract:
The vacant set is the set of vertices not visited by a random walk on a graph before a given time T. In the talk, I will discuss properties of this random subset of the graph, the phase transition conjectured in its connectivity properties (in the thermodynamic limit' when |G| and T grow simultaneously), and the relation of the problem to the random interlacement percolation. I will then concentrate on the case when G is a large-girth expander or a random regular graph, where the conjectured phase transition (and much more) can be proved.
Warren Weaver Hall Room 317 at 10:00 am.

Monday, May 2 - Special Seminar (Note the time and place)

Josef Teichmann, ETH, Zürich.
Affine Processes on Positive Semi-Definite Matrices
Abstract:
Classification and applications of affine processes on positive semi-definite matrices is presented. These processes contain OU processes on positive semi-definite matrices, and Wishart processes. Generalizations towards symmetric cones are discussed.
Warren Weaver Hall Room 1314 at 10:00 am.

Friday, May 6 - No talk today

Friday, June 3

Mark Holmes,University of Auckland.
Random walks in degenerate random environments.
Abstract:
In joint work with Tom Salisbury, we study random walks in i.i.d. random environments in Z^2 in dimensions 2 and higher. In our environments, at any given site some steps may not be available to the random walker (i.e. we don't assume ellipticity). Among our main results are 0-1 laws for directional transience (extending results already known under the assumption of ellipticity) and a simple monotonicity result for 2-valued environments (at each site the environment takes one of two values).
Warren Weaver Hall Room 317 at 10:00 am.

## Fall Semester 2010

Friday, September 24

Partha Dey, Courant Institute
Central Limit Theorem for First-Passage Percolation across thin cylinders.
Abstract:
We consider first-passage percolation on the graph ℤ×{-hn,-hn+1,...,hn}d-1 where each edge has an i.i.d. nonnegative weight. The passage time for a path is defined as the sum of weights of all the edges in that path and the first-passage time between two vertices is defined as the minimum passage time over all paths joining the two vertices. We show that the first-passage time Tn between the origin and the vertex (n,0,...,0) satisfies a Gaussian CLT as long as hn=o(nα) with α < 1/(d+1). The proof is based on moment estimates, a decomposition of Tn as an approximate sum of independent random variables and a renormalization type argument. We conjecture that the CLT holds upto hn=o(n2/3) for d=2 and provide some numerical support for that.
Joint work with Sourav Chatterjee.
Warren Weaver Hall Room 317 at 10:00 am.

Friday, October 1

Euler hydrodynamics for attractive particle systems in random environment.
Abstract:
We prove quenched hydrodynamic limit under hyperbolic time scaling for bounded attractive particle systems on ℤ in random ergodic environment. Our result is a strong law of large numbers.
Joint work with C. Bahadoran, H. Guiol, K. Ravishankar.
Warren Weaver Hall Room 317 at 10:00 am.

Friday, October 8

Tom LaGatta, Courant Institute
Riemannian First-Passage Percolation
Abstract:
Riemannian first-passage percolation is a continuum analogue of lattice FPP. Instead of considering a random metric on the lattice ℤ2, we begin with a random Riemannian metric on ℝ2. The global structures of the two models are similar - with my advisor Janek Wehr, we have proved a shape theorem for this model, which shows that balls under the Riemannian metric grow asymptotically like Euclidean balls. However, there is also a rich local structure, since Riemannian geometry provides us with notions of curvature and geodesics, curves which (locally) minimize length. Geodesics need not always globally minimize length (e.g., great circles on the sphere), and it is an interesting and important question to identify those geodesics which do so. No geometric background will be required for this talk.
Warren Weaver Hall Room 317 at 10:00 am.

Friday, October 15

Clément Hongler, Columbia University
Conformal invariance of the Ising energy field.
Abstract:
We consider the planar Ising model from a conformal invariance point of view. We are interested in the scaling limit of the model at criticality. Physics theories, notably Conformal Field Theory, predict the existence of two conformal fields underlying the model: the spin and the energy density. We have recently proved the conjectured formulae for the energy field, with an improved precision, using discrete complex analysis techniques, thanks to the introduction of holomorphic spinors. More specifically, we relate the correlation functions of the energy to special values of the spinors, and prove convergence of the latter to continuous holomorphic spinors, giving scaling formulae for the correlation functions.
Partly based on joint work with Stas Smirnov.
Warren Weaver Hall Room 317 at 10:00 am.

Friday, October 22 - Two consecutive talks.

Quasi-potential for the asymmetric exclusion process.
Abstract:
The purpose of this work is to recover by a dynamical approach the stationary large deviation functional derived by Derrida, Lebowitz & Speer (2003) for the asymmetric exclusion process in contact with reservoirs. A remarkable feature of this functional is its nonlocality, which is a signature of long-range correlations. The DLS functional is recovered and somewhat generalized by computing the quasi-potential associated to a suitable dynamical energy functional. While this approach was set up by Bertini et al. (2002) for symmetric and weakly asymmetric systems, it was so far lacking for strongly asymmetric systems, due to the different nature of the dynamical functional. The latter is a combination of a bulk functional based on entropy production (Jensen 2000, Varadhan 2004, Belletini et al. 2010) and boundary costs that measure violation from Bardos-Leroux-Nédélec boundary conditions in Burgers's equation (Bodineau & Derrida 2005).
Warren Weaver Hall Room 317 at 10:00 am.

Domokos Szász, Budapest University of Technology
Energy transfer and joint diffusion.
Abstract:
The joint diffusion of two particles in a dynamical environment was shown to become asymptotically independent for a 1-D degenerate mechanical model (Harris-Spitzer model) by the speaker in 1980, and for stochastic models of symmetric exclusion by Kipnis and Varadhan in 1985. In truly mechanical systems, however, where the interaction of the particles also involves energy exchange, this independence does not hold anymore. The phenomenon is explained and demonstrated for a stochastic model of two Lorentz disks. The diffusive limit of the motion of one particle is a mixture of Wiener processes and the random covariances are determined by the Boltzmann's Stosszahlansatz. The results are joint with Zs. Pajor-Gyulai.
Warren Weaver Hall Room 317 at 11:15 am.

Friday, October 29

Fredrik Johansson Viklund, Columbia University
Convergence rates for loop-erased random walk
Abstract:
Loop-erased random walk (LERW) is a self-avoiding random walk obtained by chronologically erasing the loops of a simple random walk. In the plane, the lattice size scaling limit of LERW is known to be SLE(2), a random fractal curve constructed by solving the Loewner differential equation with a Brownian motion input. In the talk, we will discuss recent joint work with C. Benes (CUNY) and M. Kozdron (U. of Regina) on obtaining a rate for the convergence of LERW to SLE(2). More precisely, we will outline our derivation of a rate for the convergence of the Loewner driving function for LERW to Brownian motion with speed 2 on the unit circle, the Loewner driving function for SLE(2). We will then show how to use this to obtain a rate for the convergence of the paths with respect to Hausdorff distance. Time permitting, we will also indicate how some of these results can be extended to certain other models known to converge to SLE.
Warren Weaver Hall Room 317 at 10:00 am.

Friday, November 5

Louis-Pierre Arguin, Courant Institute
Statistics of Branching Brownian Motion at the edge
Abstract:
Branching Brownian motion (BBM) is a Markov process where particles perform Brownian motion and independently split into two independent Brownian particles after an exponential holding time. The extreme value statistics of BBM in the limit of large time is of interest since BBM constitutes a borderline case, among Gaussian processes, where correlations start to affect the statistics. The law of the maximum of BBM has been understood since the works of Bramson and McKean. But little is known about the distribution of the particles close to the maximum. In this talk, I will present results on the correlation structure of these particles. This is used to unravel a Poissonian structure underlying the point process of particles at the edge. This is joint work with A. Bovier and N. Kistler.
Warren Weaver Hall Room 317 at 10:00 am.

Friday, November 12 - Two consecutive talks.

Horng-Tzer Yau, Harvard University.
Random matrices and the conjectures of Wigner and Dyson.
Abstract:
Random matrices were introduced by E. Wigner to model the excitation spectra of large nuclei. The central idea is based on the hypothesis that the local statistics of the excitation spectrum for a large complicated system is universal in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. Dyson Brownian motion is the flow of eigenvalues of random matrices when each matrix element performs independent Brownian motions. In this lecture, we will explain the connection between the universality of random matrices and the approach to local equilibrium of Dyson Brownian motion. This connection has led to a complete solution of the universality conjecture by Wigner, Dyson, Gaudin and Mehta. The main tools in our approach are an estimate on the flow of entropy in Dyson Brownian motion and a local semicircle law. One key feature of the entropy estimate is an extension of the logarithmic Sobolev inequality to cases not covered by the convexity criterion of Bakry and Emery.
Warren Weaver Hall Room 317 at 10:00 am.

Tom Alberts, University of Toronto.
Intermediate Disorder for Directed Polymers in Dimension 1+1, and the Continuum Random Polymer.
Abstract:
The 1+1 dimensional directed polymer model is a Gibbs measure on simple random walk paths of a prescribed length. The weights for the measure are determined by a random environment occupying space-time lattice sites, and the measure favors paths to which the environment gives high energy. For each inverse temperature $\beta$ the polymer is said to be in the weak disorder regime if the environment has little effect on it, and the strong disorder regime otherwise. In dimension 1+1 it turns out that all positive $\beta$ are in the strong disorder regime. I will introduce a new regime called intermediate disorder, which is accessed by scaling the inverse temperature to zero with the length $n$ of the polymer. The precise scaling is $\beta n^{-1/4}$. The most interesting result is that under this scaling the polymer has diffusive fluctuations, but the fluctuations themselves are not Gaussian. Instead they are still coupled to the random environment, and their distribution is intimately related to the Tracy-Widom distribution for the largest eigenvalue of a random matrix from the GUE. More recent work also indicates that we can take a scaling limit of the entire intermediate disorder regime to construct a continuous random path under the effect of a continuum random environment. We call the scaling limit the continuum random polymer. I will discuss a few properties of the continuum random polymer and its intimate connection to the stochastic heat equation in one dimension.
Joint work with Kostya Khanin and Jeremy Quastel.
Warren Weaver Hall Room 317 at 11:15 am.

Friday, November 18-19

Invited speakers:

Nathanael Berestycki, Cambridge University, Asymptotic behaviour of near-critical branching Brownian motion
Persi Diaconis, Stanford University, On Adding a List of Numbers (and other one-dependent determinental processes)
Yves Le Jan, Université Paris Sud, The determinant of the Green function
Edwin Perkins, University of British Columbia, Uniqueness and non-uniqueness for parabolic Stochastic PDE

Held at CUNY's Graduate Center. More details at the seminar website.

Friday, November 26

Thanksgiving holiday. No Seminar.

Friday, December 3

Shirshendu Chatterjee, Cornell University
Asymptotic Behavior of Aldous' Gossip Process.
Abstract:
Aldous (2007) defined a gossip process in which space is a discrete torus of size N, and the state of the process at time t is the set of individuals who know the information. Information spreads from a site to its nearest neighbors at rate 1/4 each and at rate N^{-\alpha} to a site chosen at random from the torus. We will be interested in the case in which α < 3, where the long range transmission significantly accelerates the time at which everyone knows the information. We prove three results that precisely describe the spread of information and asymptotic behavior of the cover time in a slightly simplified model on the (real) torus.
Warren Weaver Hall Room 317 at 10:00 am.

Friday, December 10 - Two consecutive talks.

Michael Aizenman, Princeton University
Resonant Delocalization through Large Deviations for Random Operators on Tree Graphs.
Abstract:
We resolve an existing question concerning the mobility edge for operators with a hopping term and a random potential on regular tree graphs. The model has been among the earliest studied for Anderson localization, and it continues to attract attention because of analogies with localization issues for many particle systems. A resonance mechanism is identified which causes the somewhat surprising appearance of absolutely continuous spectrum well beyond the energy band of the operator's hopping term. For weak disorder this includes a Lifshitz tail regime of very low density of states. (Joint work with S. Warzel.)
Warren Weaver Hall Room 317 at 10:00 am.

Janek Wehr, University of Arizona
Brownian motion in a diffusion gradient and exotic stochastic integrals.
Abstract:
A Brownian particle with a diffusion coefficient varying in space obeys a Newton equation of motion with a stochastic term. In the Smoluchowski-Kramers (or: overdamped) approximation, the mass of the particle is formally put equal zero, yielding a first order stochastic differential equation, which admits different interpretations, depending on the definition of the stochastic integral adopted. A recent experiment shows that the correct interpretation is "backwards Ito". I will show how this can be derived from taking the zero mass limit carefully and then discuss a whole class of similar problems. The overdamped limits can lead to equations with any definition of stochastic integration, including Ito, and backwards Ito and Stratonovitch (as a limiting case). Moreover, in a majority of these equations, the stochastic integral convention changes depending on the state of the system, going beyond Ito, Stratonovitch or any other standard definition. A series of experiments with electric circuits designed to verify these predictions is in its initial phase. This is a joint work with an experimental group in Stuttgart and with Scott Hottovy, a graduate student at the University of Arizona.
Warren Weaver Hall Room 317 at 11:15 am.

## Spring Semester 2010

Friday, January 22

Pierre Nolin, Courant Institute
Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model
Warren Weaver Hall Room 101 at 10:00 am.

Tuesday, January 26

Special Seminar

ClÈment Hongler, UniversitÈ de GenËve
Warren Weaver Hall Room 1314 at 5:00 pm.
NOTE the date and room change.

Friday, January 29

Antonio Auffinger, Courant Institute
Random Matrices and Complexity of Spin Glasses
Warren Weaver Hall Room 317 at 10:00 am.

Friday, February 5

Ivan Corwin, Courant Institute
Fluctuations of the totally asymmetric simple exclusion process
Warren Weaver Hall Room 317 at 10:00 am.

Friday, February 12

Pierluigi Falco, Institute of Advanced Study
Rigorous evaluation of critical exponents through scaling limit
Warren Weaver Hall Room 317 at 10:00 am.

Friday, February 19

Antti Kemppainen, UniversitÈ Paris-Sud and University of Helsinki
Random curves, scaling limits and Loewner evolutions
Warren Weaver Hall Room 317 at 10:00 am.

Friday, February 26

Rama Cont, Columbia University
Functional Itô calculus, integration by parts and stochastic integral representation of martingale functionals
Warren Weaver Hall Room 317 at 10:00 am.

Friday, March 5

Van Vu, Rutgers
Random matrices: Universality of the Local eigenvalues statistics
Warren Weaver Hall Room 317, at 2 pm.

Friday, March 12

Krishnamurthi Ravishankar, SUNY
Marking the Brownian web and applications
Warren Weaver Hall Room 317 at 10:00 am.

Friday, March 19

Spring Break - No Seminar.

Friday, March 26

Eyal Lubetzky, Microsoft Research
Critical slowdown for the Ising model on the two-dimensional lattice
Warren Weaver Hall Room 317 at 10:00 am.

Friday, April 2

Two Talks:

Geoffrey Grimmett, University of Cambridge
(Warren Weaver Hall Room 317 at 10:00am)

Chiranjib Mukherjee, Max Planck Institute

Tuesday, April 6

Special Seminar

G·bor Pete, University of Toronto
Room 1302, at 10:00am.
NOTE the date and room change.

Friday, April 9

S. R. Srinivasa Varadhan, Courant Institute
On some central limit theorems by Martingale Approximation
Warren Weaver Hall Room 317 at 10:00 am.

Friday, April 16

Sinan G¸nt¸rk, Courant Institute
Quantization of Random Linear Measurements
Warren Weaver Hall Room 317 at 10:00 am.

Friday, April 23

Two Talks:

Govind Menon, Brown University
Lax equations and kinetic theory for shock clustering and Burgers turbulence
Warren Weaver Hall Room 317 at 10:00 am.

Eric Nordenstam, UniversitÈ Catholique de Louvain
Seminar cancelled due to Eyjafjallajˆkull

Friday, April 30

Two Talks:

Ori Gurel-Gurevich, Microsoft Research
Poisson Thickening
Warren Weaver Hall Room 317 at 10:00 am.

Sandy Zabell, Northwestern University
Warren Weaver Hall Room 605 at 12:30 pm.

Friday, May 7

Scott Sheffield, MIT
Internal DLA and the Gaussian free field
Warren Weaver Hall Room 317 at 10:00 am.

Friday, May 14

Eric Nordenstam, UniversitÈ Catholique de Louvain
A particle dynamics related to the shuffling algorithm for the Aztec diamond
Warren Weaver Hall Room 317 at 10:00 am.

## Fall Semester 2009

Wednesdays and Fridays, September 9-30

Minerva Research Foundation Lectures at Columbia University
Jean Bertoin, Universite Paris VI
Exchangeable Coalescents
Wednesdays 10am-noon (Math 622) and Fridays 10am-noon (Math 507)
2990 Broadway at 117'th st., Columbia University

Friday, September 11

Mihyun Kang, Technische Universit‰t Berlin
Two critical behaviour of random planar graphs
Warren Weaver Hall Room 412 at 10:10 am.

Applied Mathematics Seminar talk
Patrick Dondl, Bonn University
Pinning of interfaces in random media
Warren Weaver Hall Rm 1302 at 2:30pm.

Friday, September 18

Random matrices on compact groups and independence
Warren Weaver Hall Room 517 at 12:00 noon.

Friday, September 25

Olivier Bernardi, MIT
Bijective approach to tree-rooted maps
Warren Weaver Hall Room 102 at 10:00 am.

Friday, October 2

Sourav Chatterjee, NYU and UC Berkeley
Superconcentration
Warren Weaver Hall Room 517 at 10:00 am.

Friday, October 9

Francesco Russo, INRIA Rocquencourt, Projet MATHFI and UniversitÈ Paris 13
Probabilistic representation of a generalized porous media type equation and related fields
Warren Weaver Hall Room 517 at 10:00 am.

Friday, October 16

Jian Ding, University of California, Berkeley
Near-critical random graph: its structure, diameter and mixing time
Warren Weaver Hall Room 517 at 10:00 am.

Informal Lunchtime Seminar (bring your lunch!)
Mark Meckes, Case Western Reserve University
Concentration of polynomials in random matrices
Warren Weaver Hall Room 1314 at 12:10 - 1:10pm.

Friday, October 23

Alexander Fribergh, NYU
Biased random walks on a percolation cluster
Warren Weaver Hall Room 517 at 10:00 am.

Friday, October 30

Ron Peled, NYU
High-dimensional homomorphism height functions are flat
Warren Weaver Hall Room 517 at 10:00 am.

Friday, November 6
Two Seminars

Oren Louidor, NYU
Finite connections for supercritical Bernoulli bond percolation in 2D.

Partha Dey, University of California, Berkeley
Stein's method and large deviation for number of triangles in ErdösñRÈnyi Random Graph
Both seminars are at Warren Weaver Hall Room 517, starting at 10:00 am.

Friday, November 13

Mark Kelbert, Swansea University
The branching diffusion on hyperbolic space
Warren Weaver Hall Room 517 at 10:00 am.

Thursday and Friday, November 19-20

Invited Speakers:
Rick Kenyon, Brown University
Claudia Neuhauser, University of Minnesota
Giovanni Peccati, UniversitÈ Paris Ouest
Craig Tracy, University of California, Davis

Held at the C.P. Davis Auditorium in the Schapiro Center at Columbia University.
More details at the seminar website.

Friday, November 27

Thanksgiving Holiday - No Seminar.

Friday, December 4

Lorenzo Zambotti, UniversitÈ Paris VI
An entropic functional on families of random variables from theoretical biology
Warren Weaver Hall Room 517 at 10:00 am.

Special Seminar
Fredrik Johansson, KTH
Behavior of the SLE path at the tip
Warren Weaver Hall Room 517 at 2:00 pm.

Friday, December 11

Rob van den Berg, Vrije Universiteit and CWI
Sharpness of percolation transitions in some dependent two-dimensional models
Warren Weaver Hall Room 517 at 10:00 am.

## Spring Semester 2009

Friday, January 30

No seminar scheduled to allow people to hear Andrei Okounkov speak on "Random surfaces and Algebraic curves" at Columbia.
Lecture is at 9:30am, 520 Math, Columbia university.
Okounkov's talk will be followed by Thierry Bodineau's talk on "Large deviations for non-equilibrium particle systems" in Columbia's probability seminar.
12:00 noon, 903 SSW Bldg (1255 Amsterdam Avenue-btwn. 121st & 122nd Street).

Friday, February 6

Todd Kemp, MIT
Warren Weaver Hall Room 312 at 10:00 am.

Friday, February 13

Michael Damron, Courant Institute
Michael was ill and instead Ron Peled from Courant Insitute talked about "Translation-equivariant colorings of Poisson-Voronoi diagrams".
Warren Weaver Hall Room 312 at 10:00am.

Friday, February 20

Sasha Sodin, Tel-Aviv University
Warren Weaver Hall Room 312 at 10:00 am.

Monday, February 23 and Wednesday, February 25

Special lectures in probability at Columbia University

Etienne Pardoux, Marseille

First talk is on Monday, February 23, 9:30-11:00am, Hamilton 517, Columbia University.
"Can a single mutant's progeny survive for ever?"
Second talk is on Wednesday, February 25, 9:30-11:00am, SSW 1025, Columbia University.
"'Homegenization and SPDE's"

Note the unusual place and time! More details at the Columbia Probability Seminar website.

Friday, February 27

Dmitry Panchenko, Texas A&M
(Warren Weaver Hall Room 312 at 10:00am)

Friday, March 6

Special Joint Columbia / Courant Seminar

Yuval Peres, Microsoft Research
(Warren Weaver Hall Room 312 at 10:00am)

Note also Yuval Peres' talk at the Courant Institute Mathematics Colloquium on Monday, March 2'nd.

Friday, March 13

Vincent Vargas, Université Paris Dauphine
(Warren Weaver Hall Room 312 at 10:00am)

Friday, March 20

Spring Break - No seminar scheduled

Friday, March 27

Kay Kirkpatrick, MIT
(Warren Weaver Hall Room 312 at 10:00am)

Friday, April 3

Martin Hairer, Courant Institute
(Warren Weaver Hall Room 312 at 10:00am)

Friday, April 10

Lionel Levine, MIT
(Warren Weaver Hall Room 312 at 10:00am)

Friday, April 17

Two Talks:

Xue-Mei Li, University of Warwick
(Warren Weaver Hall Room 312 at 10:00am)

Charles Radin, University of Texas at Austin
"Modeling Sand"

Friday, April 24

Seminar Cancelled.

Monday, April 27

Special Seminar

Federico Camia, Vrije Universiteit
Warren Weaver Hall Room 1302, at 10:00am.
NOTE the date and room change.

Wednesday, April 29

Special Seminar

Horng-Tzer Yau, Harvard University
Warren Weaver Hall Room 512 at 11:30am.
NOTE the date and room change.

Friday, May 1

Columbia-Princeton Probability Day

Held at Columbia University, Schermerhorn Hall room 501.
Registration/coffee from 9-10am, lectures follow.

Full program and map here.

The NYU spring semester ends on Monday, May 4'th. Some talks will be held during the summer, see below.

Friday, May 15

Yuri Kifer, Hebrew University of Jerusalem
(Warren Weaver Hall Room 312 at 10:00am)

Thursday, July 2

ClÈment Hongler, UniversitÈ de GenËve
Warren Weaver Hall Room 1314 at 11:00am.
NOTE the date and room change.

## Fall Semester 2008

Friday, November 7

Thierry Bodineau, École Normale Supérieure
"Current large deviations in stochastic systems"
(Warren Weaver Hall Room 312 at 10:00 am)

Friday, October 24

Joint Columbia/Courant seminar (note the unusual place and time)

Brian Rider, University of Colorado, Boulder
"Beta Ensembles, Random Schroedinger, and Diffusion"
(Room 507 at Columbia at 10:45 am)

George Papanicolaou, Stanford University
"Modeling fine-scale uncertainty in Bayesian parameter estimation and applications"
(Room 520 at Columbia at 12:00 pm)

Friday, October 17

Michael Damron, Courant Institute
"Invasion percolation in 2D"
(Warren Weaver Hall Room 312 at 10:00 am)

Friday, October 10

José Ramírez, Universidad de Costa Rica
"Diffusion limits for eigenvalues of random matrices"
(Warren Weaver Hall Room 312 at 10:00 am)

Friday, October 3

Pierre Nolin, Courant Institute
"A particular bit of universality: inhomogeneity and SLE(6)"
(Warren Weaver Hall Room 312 at 10:00 am)

Friday, September 26

Ofer Zeitouni, Weizmann Institute & University of Minnesota
"Exit measures for isotropic Random walk in random environments - a perturbative approach"
(Warren Weaver Hall Room 312 at 10:00 am)

Nina Gantert, Universität Münster
"Survival time of random walk in random environment among soft obstacles"
(Warren Weaver Hall Room 312 at 11:15 am)

## Abstracts

### Spring 2010

Pierre Nolin, Courant Institute
Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model .
Abstract: For two-dimensional independent percolation, Russo-Seymour-Welsh (RSW) bounds on crossing probabilities are an important a-priori indication of scale invariance, and they turned out to be instrumental to describe the phase transition. They are in particular a key tool to derive the so-called scaling relations, that link the critical exponents associated with the main macroscopic functions.

In this talk, we prove RSW-type uniform bounds on crossing probabilities for the FK Ising model at criticality, independent of the boundary conditions. A central tool in our proof is Smirnov's fermionic observable for the FK Ising model, that makes appear some harmonicity on the discrete level, providing precise estimates on boundary connection probabilities. We also prove several related results - some new and some not - among which the fact that there is no magnetization at criticality, tightness properties for the interfaces, and the value of the half-plane one-arm exponent.

This is joint work with H. Duminil-Copin and C. Hongler.

ClÈment Hongler, UniversitÈ de GenËve
Convergence of Ising model interfaces to dipolar SLE.
Abstract: We consider the interfaces of the critical planar Ising model on the square lattice. In a Dobrushin setup, that is, when the boundary conditions are [+] on a boundary arc and [-] on the rest, the interface between [+] and [-] spins has been shown by Smirnov (and Chelkak-Smirnov for more general lattices) to converge to chordal SLE(3).
The three types of boundary conditions of the Ising model that have been predicted to be conformally invariant are [+], [-] and free, as well as combinations of them. In the case of dipolar boundary conditions, that is, when the boundary is split into [+], [-] and free arcs, the interface starting between [+] and [-] has been conjecture by physicists to converge to a variant of SLE, called dipolar SLE(3), thus generalizing Smirnov's result.
We will give the proof of this conjecture. It relies on the introduction of a new martingale observable, which plays the role of a stochastically conserved quantity, and helps deducing conformal symmetry of the model.
The introduction of the martingale observable is made through the introduction of a dual model. Using a remarkable combinatorial identity, known as Kramers-Wannier duality, we prove that obtaining a martingale observable for the interface can be made by understanding spin-spin correlations on a dual Ising model.
Using the FK representation of this dual Ising model and the scaling limit of its interfaces which is SLE(16/3), as well as the convergence of the discrete holomorphic fermions introduced by Smirnov for the FK and the spin Ising models, we manage to express these spin-spin correlations as SLE integrals. These integrals are finally computed using Conformal Field Theory-inspired computations (relying notably on solutions of Dotsenko-Fateev equations).
Our method is in fairly general and allows in principle to identify the scaling limit of interfaces in all the conformally invariant boundary conditions setups. It can moreover be used to prove early predictions about crossing probabilities in the Ising model and is the starting point of the construction of a free boundary conditions version of the Conformal Loop Ensembles.

Antonio Auffinger, Courant Institute
Random Matrices and Complexity of Spin Glasses.
Abstract: We introduce a new identity, relating random matrix theory and the problem of counting the number of critical points of certain random (Gaussian) functions in high dimensional spheres, the Hamiltonians of spherical spin-glass models. This identity allows us to describe an interesting layered structure of local minima and saddle points at low levels of energy and to compute the ground state energy of these Hamiltonians.

This is joint work with G. Ben Arous (Courant) and J. Cerny (ETHZ).

Ivan Corwin, Courant Institute
Fluctuations of the totally asymmetric simple exclusion process.
Abstract: We study how the evolution of this process fluctuates around its expected behavior. For TASEP started with two-sided Bernoulli initial conditions we provide a complete characterization of the limiting (large time) fluctuation processes. These processes vary according to the region in the hydrodynamic limit. Results proved can be interpreted also in terms of last passage percolation, crystal growth models, queues in series, and spiked Wishart random matrices.

This includes joint with GÈrard Ben Arous, and with Patrik Ferrari and Sandrine PÈchÈ.

Pierluigi Falco, Institute of Advanced Study
Rigorous evaluation of critical exponents through scaling limit.
Abstract: I will introduce some critical features of the Eight-Vertex and the Ashkin-Teller models; and I will discuss how the use of Renormalization Group permits the rigorous proof of some scaling formulas conjectured by Kadanoff. In the Eight-Vertex case, these formulas give the exact values of some critical exponents that are not computable through the Baxter's exact solution.

Antti Kemppainen, UniversitÈ Paris-Sud and University of Helsinki
Random curves, scaling limits and Loewner evolutions.
Abstract: In the 2D statistical physics and its lattice models, interfaces are random curves. A general method to prove the convergence of a random discrete curve, as the lattice mesh goes to zero, is to first show the existence of subsequent scaling limits and then to prove the uniqueness. In this talk, I will introduce a sufficient condition, and some equivalent formulations, that guarantee the precompactness (existence) and also that the limits are Loewner evolutions, i.e. they correspond to continuous Loewner driving processes. The second result is needed for the unique characterization of the limits. This framework of estimates can be used for almost all of the already existing proofs of an interface converging to a Schramm-Loewner evolution (SLE), and for at least one new result. In principle, it can be applied beyond SLE.

Joint work with Stanislav Smirnov, UniversitÈ de GenËve

Rama Cont, Columbia University
Functional Itô calculus, integration by parts and stochastic integral representation of martingale functionals.
Abstract: We develop a non-anticipative calculus for functionals of a continuous semimartingale, using a pathwise functional derivative recently proposed by B Dupire. A functional extension of the Ito formula is derived, and used to obtain a constructive martingale representation theorem for a class of continuous martingales verifying a regularity property. By contrast with the Clark-Ocone formula, this representation involves non-anticipative quantities which may be computed pathwise. The martingale representation formula allows to obtain an integration by parts formula for Ito stochastic integrals, which in turn enables to define a non-anticipative weak functional derivative for a class of square-integrable martingales. We show that this weak derivative is the adjoint of Ito stochastic integral and may be viewed as a non-anticipative lifting" of the Malliavin derivative. Finally, regular functionals of an Ito martingale which have the local martingale property are characterized as solutions of a functional differential equation, for which a uniqueness result is given.

Joint work with: David FOURNIE (Columbia University).

Van Vu, Rutgers
Random matrices: Universality of the Local eigenvalues statistics.
Abstract: One of the main goals of the theory of random matrices is to establish the limiting distributions of the eigenvalues. In the 1950s, Wigner proved his famous semi-cirle law (subsequently extended by Anord, Pastur and others), which established the global distribution of the eigenvalues of random Hermitian matrices.
In the last fifty years or so, the focus of the theory has been on the local distributions, such as the distribution of the gaps between consecutive eigenvalues, the k-point correlations, the local fluctuation of a particular eigenvalue, or the distribution of the least singular value. Many of these problems have connections to other fields of mathematics, such as combinatorics, number theory, statistics and numerical linear algebra.
Most of the local statistics can be computed explicitly for random matrices with gaussian entries (GUE or GOE), thanks to Ginibre's formulae of the joint density of eigenvalues. It has been conjectured that in the limit the same results hold for other models of random matrices. This is generally known as the Universality phenomenon, which has been supported by overwhelming numerical evidence and various concrete conjectures.
In this talk, we would like to discuss recent progresses concerning the Universality phenomenon, focusing on a recent result (obtained jointly with T. Tao), which asserts that all local statistics of eigenvalues of a random matrix are determined by the first four moments of the entries. This provides the answer to several old problems.
The method also extends to other models of random matrices, such as sample covariance matrices.

Krishnamurthi Ravishankar, SUNY
Marking the Brownian web and applications
Abstract

Eyal Lubetzky, Microsoft Research
Critical slowdown for the Ising model on the two-dimensional lattice.
Abstract: Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the Glauber dynamics for the Ising model on $Z^2$ everywhere except at criticality. While the critical behavior of the Ising model has long been the focus for physicists, mathematicians have only recently developed an understanding of its critical geometry with the advent of SLE, CLE and new tools to study conformally invariant systems.
A rich interplay exists between the static and dynamic models. At the static phase-transition for Ising, the dynamics is conjectured to undergo a critical slowdown: At high temperature the inverse-gap is $O(1)$, at the critical $\beta_c$ it is polynomial in the side-length and at low temperature it is exponential in it. A long series of papers verified this on $Z^2$ except at $\beta=\beta_c$ where the behavior remained unknown.
In this work we establish the first rigorous polynomial upper bound for the critical mixing, thus confirming the critical slowdown for the Ising model in $Z^2$. Namely, we show that on a finite box with arbitrary (e.g. fixed, free, periodic) boundary conditions, the inverse-gap at $\beta=\beta_c$ is polynomial in the side-length. The proof harnesses recent understanding of the scaling limit of critical Fortuin-Kasteleyn representation of the Ising model together with classical tools from the analysis of Markov chains.

Based on joint work with Allan Sly.

Geoffrey Grimmett, University of Cambridge
Embeddings, entanglement, and percolation.
Abstract: Can there exist a monotone embedding of one infinite random word inside another, with bounded gaps? What can be said about the critical point for the existence of an infinite entangled' set of open edges in the percolation model on the cubic lattice?
These two questions are connected through a new type of percolation process, called Lipschitz percolation'. It will be shown (amongst other things) how to embed some higher-dimensional words, and to obtain the best (so far) lower bound for the entanglement critical point.

This work is joint with Ander Holroyd, and has benefited from the Courant Institute seminar series.

Chiranjib Mukherjee, Max Planck Institute
Brownian intersection local times and large deviations.
Abstract: We consider a number of independent Brownian motions running in the d-dimensional Euclidean space until they exit a fixed ball. We look at the spatial intersection of the paths. Le Gall and others constructed an object measuring the intensity of the path intersections in the set. Keeping track of the notion of local time pertaining to a single path, this object is called the Brownian intersection local time''. Koenig and Moerters recently studied the upper tails of this random object, sending the amount of intersection on a fixed compact set to infinity. The resulting variational formula admits minimizer(s) with certain probabilistic interpretation along the line of classical Donsker-Varadhan theory. Inspired by this, we study large deviations for normalised intersection local times (as a measure) for a fixed time horizon in the ball. As a corollary to this, we obtain an LDP for normalised intersection local times, for motions observed until individual exit times.

G·bor Pete, University of Toronto
Random walk on percolation clusters, and scale-invariant groups.
Abstract: There are well-known connections between geometric properties of Cayley graphs and the behavior of simple random walk on them. But most tools stop working if we consider random walk inside an infinite percolation cluster of the graph, even though the same results should hold.
In the first part of the talk, I give a *simple* proof that the isoperimetric profile of the infinite cluster basically coincides with the profile of the lattice Z^d for any percolation density p>p_c(Z^d), and for p close enough to 1 on Cayley graphs of finitely presented groups. This implies that the on-diagonal heat kernel decay survives percolation.
The situation on Z^d is better than in general because of a standard percolation technique called renormalization. So, in the second part, I will examine the possibility of renormalization on other Cayley graphs. A group G is called scale-invariant if it has a nested chain of finite index subgroups, all isomorphic to G, whose intersection is trivial. Itai Benjamini conjectured that scale-invariant groups must have polynomial volume growth. In joint work with V. Nekrashevych, I have given several counterexamples, including the lamplighter group Z_2 \wr Z.
I will give a lot of open questions.

S. R. Srinivasa Varadhan, Courant Institute
On some central limit theorems by Martingale Approximation.
Abstract: We will investigate the CLT for sums of the form $\sum_{i=1}^n f(X_i, X_{2i},\ldots, X_{ki})$ where $\{X_i\}$ are dependent random variables with some mixing properties.

Sinan G¸nt¸rk, Courant Institute
Quantization of Random Linear Measurements.
Abstract: We will discuss the problem of how to quantize m random linear measurements of k-dimensional vectors, where m > k. The standard choice is to round each measurement vector to the nearest lattice point, and to reconstruct via the (Moore-Penrose) pseudo-inverse. This talk is about a quantization and reconstruction alternative which relies on the concept of "noise-shaping" in analog-to-digital conversion, Sobolev-dual frames, and concentration of singular values for certain families of random matrices. We will also present implications and improvements for compressed sensing.

Joint work with M. Lammers, A. Powell, R. Saab, and O. Yilmaz.

Govind Menon, Brown University
Lax equations and kinetic theory for shock clustering and Burgers turbulence.
Abstract: Much of our current understanding of statistical theories of turbulence relies on vastly simplified caricatures. One such caricature is Burgers turbulence. This is the study of the statistics of shocks in Burgers equation with random initial data or forcing. This model also arises in statistics, combinatorics, and models of coagulation and surface growth. It is of wide interest as a benchmark, even if it describes phenomena that are not entirely turbulent.
I will describe a kinetic theory for shock clustering that applies to all scalar conservation laws with convex flux. A remarkable feature of the kinetic theory is that it is presented as a Lax pair, admits surprising exact solutions, and has intriguinging connections with completely integrable systems and random matrix theory. This is mostly joint work with Bob Pego (CMU) and Ravi Srinivasan (UT, Austin).

Eric Nordenstam, UniversitÈ Catholique de Louvain
A particle dynamics related to the shuffling algorithm for the Aztec diamond.
Abstract: The shuffling algorithm (introduced by Elkies et al.) for sampling a tiling of the Aztec diamond uniformly at random can be seen as a certain dynamics on a set of interacting particles. This is a discretisation of a model of interlaced Brownian motions recently studied by Warren. As an application of these results, I will sketch a new proof of that fact that, in a suitable scaling limit of large Aztec diamonds, one can recover the distribution of the eigenvalues of a GUE matrix and its principal minors.

This work is related to recent work of Borodin and Ferrari.

Ori Gurel-Gurevich, Microsoft Research
Poisson Thickening.
Abstract: Can a Poisson process be thickened? That is, can more points be added deterministically to a Poisson process, so that the resulting process is also a Poisson process (of higher intensity)? We will show that this can be done, but not equivariantly (i.e. not in a way which commutes with some shift).
In recent years, there has been much interest in problems of this kind: given a stochastic spatial process X, can it be extended to another process Y (perhaps under additional constraints)? For example, can the cells of a Poisson-Voronoi tessellation be colored deterministically and equivariantly, such that adjacent cells have different colors?
We will survey results of this kind, with particular emphasis on those which yield pretty pictures and explain the solution to the thickening problem in some detail.

Joint Work with Ron Peled.

Sandy Zabell, Northwestern University
A large deviation result for pinned random walks with barrier curves
Abstract

Scott Sheffield, MIT
Internal DLA and the Gaussian free field.
Abstract: Internal diffusion limited aggregation (DLA) is a simple and natural random growth model with a beautiful history. I will describe some recent work joint with David Jerison and Lionel Levine on this subject. This work includes a proof of the "logarithmic-fluctuation" conjecture. It also precisely describes the scaling limit of the random fluctuations. The Gaussian free field makes a surprise appearance.

### Fall 2009

Mihyun Kang, Technische Universit‰t Berlin
Two critical behaviour of random planar graphs
Abstract

Random matrices on compact groups and independence.
Abstract: The Chinese restaurant process gives an iterative construction of the Ewens measures on the symmetric group. We will apply this idea to any unitary group, generating in particular its Haar measure by composing independent reflections. As a consequence, for a random matrix uniformly distributed on a compact group, the characteristic polynomial is a product of independent random variables. We will also explain how these results are linked to classical number-theoretic conjectures.

Olivier Bernardi, MIT
Bijective approach to tree-rooted maps
Abstract: Planar maps are connected planar graphs embedded in the 2-dimensional sphere, and considered up to homeomorphisms. These objects are of interest, in particular, as models of random geometries. Many recent advances in the theory of maps are based on bijections between maps and certain decorated plane trees.
In this talk, I will consider "tree-rooted maps", that is, maps with a marked spanning tree. I will present a bijection between tree-rooted maps and pairs of plane trees. I will explain the link between this bijection and a bijection by Schaeffer/Bouttier-Di Francesco-Guitter which is fundamental for studying the metric properties of maps. Lastly, I will present a generalization of the bijection to orientable surfaces other than the sphere and its enumerative consequences.

Sourav Chatterjee, NYU and UC Berkeley
Superconcentration
Abstract: We introduce the term superconcentration' to describe the phenomenon when a function of a Gaussian random field exhibits a far stronger concentration than predicted by classical concentration of measure. We show that when superconcentration happens, the field becomes chaotic under small perturbations and a multiple valley picture' emerges. Conversely, chaos implies superconcentration. While a few notable examples of superconcentrated functions already exist, e.g. the largest eigenvalue of a GUE matrix, we show that the phenomenon is widespread in physical models; for example, superconcentration is present in the Sherrington-Kirkpatrick model of spin glasses, directed polymers in random environment, the Gaussian free field and the Kauffman-Levin model of evolutionary biology. As a consequence we resolve the long-standing physics conjectures of disorder-chaos and multiple valleys in the Sherrington-Kirkpatrick model, which is one of the focal points of this talk.

Francesco Russo, INRIA Rocquencourt, Projet MATHFI and UniversitÈ Paris 13
Probabilistic representation of a generalized porous media type equation and related fields
Abstract

Jian Ding, University of California, Berkeley
Near-critical random graph: its structure, diameter and mixing time
Abstract

Mark Meckes, Case Western Reserve University
Concentration of polynomials in random matrices
Abstract:In the spirit of results of Guionnet and Zeitouni and of free probability theory, we prove concentration inequalities for noncommutative polynomials of large independent random matrices. This is joint work with S. Szarek.

Alexander Fribergh, NYU
Biased random walks on a percolation cluster
Abstract:We will present a model of random walk in random environments (RWRE) called biased random walks on a percolation cluster. This model arises from the physics literature and exhibits an unexpected slowdown phenomenon, the asymptotic speed of the random walk may actually decrease as the bias is increased. We will describe this phenomenon, how it arises and describe many open questions related to it. We will then explain how one can understand the speed of the walk on a percolation cluster of high density (p close to 1).

Ron Peled, NYU
High-dimensional homomorphism height functions are flat
Abstract:A homomorphism height function on a graph G is an integer-valued function on the vertices of G which differs by exactly one across every edge of G. One is concerned with the properties of the typical height function, that is, a function sampled uniformly among all height functions which equal 0 at some fixed point. This is a generalization of simple random walk - the case when G is a path. We take G to be a d-dimensional torus. In this case, height functions correspond to proper 3-colorings, at least for certain boundary conditions. Our main result is that in high enough dimensions, the typical height function is very flat, having bounded height at any fixed vertex and small global fluctuations. Indeed, we obtain a structure theorem for the typical function showing that it is almost constant on either the even or odd sublattices of the torus, with precise estimates for the size of breakups of this pattern. This extends results of Kahn and Galvin for the case that G is the hypercube.
Using an observation of Yadin, the results extend also to the related class of 1-Lipschitz functions on G. In addition, some information is provided on the two dimensional torus case hinting that it undergoes a certain roughening transition. This refutes a conjecture of Benjamini, Yadin and Yehudayoff.

Oren Louidor, NYU
Finite connections for supercritical Bernoulli bond percolation in 2D
Abstract:Two vertices are said to be finitely connected if they belong to the same cluster and this cluster is finite. We derive sharp asymptotics for finite connection probabilities for supercritical Bernoulli bond percolation on Z^2.

Partha Dey, University of California, Berkeley
Stein's method and large deviation for number of triangles in ErdösñRÈnyi model Random Graph
Abstract:Stein's method is a semi-classical tool for establishing distributional convergence, particularly effective in problems involving complex dependencies. A general way of deriving concentration inequalities using Stein's method was introduced by Sourav Chatterjee in 2005. Here we show how this method can be used to derive exact large deviation asymptotics for the number of triangles in the ErdösñRÈnyi Random Graph G(n,p) when p>=0.31. The proof is based on a rigorous analysis of the exponential random graph model using Stein's method for exchangeable pair. The same idea can be extended to find large deviation rate function for number of small subgraphs in G(n,p) for p above a threshold. This talk is based on joint work with Sourav Chatterjee.

Mark Kelbert, Swansea University
The branching diffusion on hyperbolic space
Abstract: We say that a branching diffusion (BD) on a Riemannian manifold $M$ is recurrent if at least one offspring of a single particle starting from $x\in M$ will return to any neighborhood of point $x$ with probability 1, and transient otherwise. The sufficient conditions for recurrency and transiency of BD are presented. For a transient BD on a hyperbolic space with a variable fission rate the Hausdorff dimension of the attractor on the absolute is evaluated.

Lorenzo Zambotti, UniversitÈ Paris VI
An entropic functional on families of random variables from theoretical biology
Abstract: G. Edelman, O. Sporns and G. Tononi have introduced in theoretical biology the neural complexity of a family of random variables, defining it as a specific average of mutual information over subsystems. We provide a mathematical framework for this concept, studying in particular the problem of maximization of such functionals for fixed system size and the asymptotic properties of maximizers as the system size goes to infinity. (Joint work with Jerome Buzzi)

Fredrik Johansson, KTH
Behavior of the SLE path at the tip
Abstract: The Schramm-Loewner evolution (SLE) is a family of random fractal curves that arise as scaling limits of two-dimensional lattice models from statistical physics. In the talk I will discuss a derivation of the optimal Holder exponent for the SLE path (in the capacity parameterization) and, briefly, a related result on the decay of harmonic measure at the tip. This is joint work with G. Lawler (University of Chicago).

Rob van den Berg, Vrije Universiteit and CWI
Sharpness of percolation transitions in some dependent two-dimensional models
Abstract: Ordinary (independent) percolation models have a sharp percolation transition: below the percolation threshold the cluster size distribution has exponential decay. For 2-dimensional models this was first proved by Kesten (1980).
In 1981 Russo proved a so-called approximate zero-one law and pointed out that a key step in Kesten's argument can be seen as a special case of this more general law. A few years ago, new results by Bollobas and Riordan for the two-dimensional Voroinoi percolation model triggered more research in that direction.
I will mainly focus on the contact process, a mathematical model of spatial epidemics, vegetation patterns and other natural random spatial structures.

### Spring 2009

Todd Kemp, MIT
Resolvents of $R$-Diagonal Ensembles
Abstract: Random matrix theory, a very young subject, studies the behaviour of the eigenvalues of matrices with random entries (with specified correlations). When all entries are independent (the simplest interesting assumption), a universal law emerges: essentially regardless of the laws of the entries, the eigenvalues become uniformly distributed in the unit disc as the matrix size increases. This circular law was first proved, with strong assumptions, in the 1980s; the current state of the art, due to Tao and Vu, with very weak assumptions, is less than a year old. It is the universality of the law that is of key interest.
What if the entries are not independent? Of course, much more complex behaviour is possible in general. In the 1990s, "$R$-diagonal" matrix ensembles were introduced; they form a large class of non-normal random matrices with (typically) non-independent entries. In the last decade, they have found many uses in operator theory and free probability; most notably, they feature prominently in Haagerup's recent work towards proving the invariant subspace conjecture.
In this lecture, I will discuss my recent joint work with Haagerup and Speicher, where we prove a universal law for the resolvent of any $R$-diagonal operator. The circular ensemble is an important special case. The rate of blow-up is, in fact, universal among all $R$-diagonal operators, with a constant depending only on their fourth moment. The proof intertwines both complex analysis and combinatorics.

Michael Damron, Courant Institute
Two Dimensional Invasion Percolation and Incipient Infinite Clusters
Abstract: In this talk, we will examine the structure of the two dimensional invasion percolation cluster (IPC) of the origin. We will review recent results about the sizes of the ponds and talk about their relation to multiple-armed generalizations of Kesten's incipient infinite cluster (IIC). In addition we will give the ideas of the proof of mutual singularity of the IPC and IIC measures.
This is joint work with Artem Sapozhnikov and Balint Vagvolgyi.

Sasha Sodin, Tel-Aviv University
Universality at the edge of the spectrum for random matrices with independent entries: Soshnikov's theorems and some extensions
Abstract: We shall discuss the distribution of extreme eigenvalues for several classes of random matrices with independent entries. In particular, we shall discuss the results of Soshnikov and some of their recent extensions, and the combinatorial questions that appear in the proofs.
Based on joint work with Ohad Feldheim.

Dmitry Panchenko, Texas A&M
The Ghirlanda-Guerra identities and ultrametricity in the Sherrington-Kirkpatrick model
Abstract: The Parisi theory of the SK model completely describes the geometry of the Gibbs sample in a sense that it predicts the limiting joint distribution of all scalar products, or overlaps, between i.i.d. replicas. One of the main properties of this distribution is the ultrametricity which means that the Gibbs measure approximately concentrates on the ultrametric subset of all configurations; another property is the Ghirlanda-Guerra distributional identities. It is well known that these two properties completely determine the distribution and, probably for this reason, they were always considered complementary. We show that if in the limit an overlap takes finitely many values then the Ghirlanda-Guerra identities actually imply ultrametricity.

Yuval Peres, Microsoft Research
Is the critical percolation probability local?
Abstract: We show that the critical probability for percolation on a d-regular non-amenable graph of large girth is close to the critical probability for percolation on an infinite d-regular tree. This is a special case of a conjecture due to O. Schramm on the locality of p_c. We also prove a finite analogue of the conjecture for expander graphs.
Joint work with Itai Benjamini and Asaf Nachmias.

Vincent Vargas, Université Paris Dauphine
Stochastic Scale Invariance and the KPZ formula
Abstract: In this talk, we will prove the KPZ equation (initially introduced in the framework of quantum gravity in 2 dimensions) for the limit lognormal measures introduced by Mandelbrot. More specifically, for a given set K, we will relate it's Hausdorff dimension under the Euclidian metric to it's Hausdorff dimension under the random metric induced by the limit lognormal measure. We will see how the notion of stochastic scale invariance is crucial in the derivation of the aforementioned relation. (Joint work with R. Rhodes)

Kay Kirkpatrick, MIT
Quantum many-body systems and the nonlinear Schroedinger equation
Abstract: At extremely cold temperatures there forms a new state of matter, called Bose-Einstein condensation, with weird behavior: quantum effects are visible macroscopically, and friction no longer matters. Certain aspects of this phenomenon are nicely understood by scaling limits.

We describe two scaling limits for systems of many quantum particles: mean-field systems and Bose-Einstein condensation. First, in mean-field systems, the microscopic particles experience weak and diffuse interactions, and the Hartree equation provides the macroscopic description. Second, in Bose-Einstein condensation (which can be viewed as a limiting case of mean-field systems), the particles experience strong and short-scale interactions, and the cubic nonlinear Schroedinger equation provides the macroscopic description.

In recent joint work with Benjamin Schlein and Gigliola Staffilani, we have handled the two-dimensional Bose-Einstein condensation--and the periodic case is especially interesting, as it involves some techniques from analytic number theory.

Martin Hairer, Courant Institute
A 'weak convergence' alternative to Harris chains
Abstract: One of the most commonly used theories to prove (strong) convergence of a Markov chain to its invariant measure is the theory of Harris chains. One major drawback of this theory is that it requires a lower bound on transition probabilities, which fails to hold in many infinite-dimensional examples where transition probabilities are mutually singular. This is the case for example for some stochastic PDEs, as well as some stochastic delay equations.
We provide an alternative theory which allows to obtain constructive criteria for weak convergence, thus exploiting the topology of the state space. In particular, we obtain a "weak form" of Harris's theorem, which yields spectral gap results in Wasserstein-type distances. These results are also of interest in the finite-dimensional case as they yield simple stability theorems for the invariant measure under weak approximations of the semigroup.

Lionel Levine, MIT
Growth rates and explosions in sandpiles
Abstract: The abelian sandpile model in Z^d produces beautiful examples of pattern formation, most of which are not yet well understood. I'll discuss a pair of conjectures about the scale invariance and dimensional reduction of the patterns formed. I'll also explore the dichotomy between robust and explosive sandpile configurations. The former are configurations to which adding a finite amount of additional sand produces only finitely many topplings. An example is the constant configuration of 2 chips at each site in Z^2. We prove a "least action principle" and use it to bound the diameter of the set of sites that topple. If an arbitrarily small fraction of sites chosen at random start with 3 chips instead of 2, however, the result is an explosion: every site in Z^2 topples infinitely often.
Joint work with Anne Fey and Yuval Peres.

Xue-Mei Li, University of Warwick
A negative result for Stochastic Differential Equations
Abstract: The solution to an ordinary differential equation depends on its initial data continuously provided that it has a global solution. This is not the case for stochastic differential equations. Positive results have been searched for long and hard. For a global strong solution to exist, the vector fields should have linear growth at infinity (in the forward direction), allowing logarithmic order corrections. The regularity needed for the vector fields are locally Lipschtz. The question is how to construct examples of conservative SDEs which has no global smooth solutions. The counter examples we knew so far do not satisfy the linear grwoth condition.
We construct a SDE without a global smooth flow whose coefficients are bounded and smooth. Only finite dimensional noise is needed. This is joint work with M. Scheutzow.

Federico Camia, Vrije Universiteit
Ising(Conformal) Fields and Cluster Area Measures
Abstract: I will discuss a representation for the magnetization field of the critical two-dimensional Ising model in the scaling limit as a random field using renormalized area measures associated with SLE clusters. The renormalized areas come from the scaling limit of critical FK (Fortuin-Kasteleyn) clusters and the random field is a convergent sum of the area measures with random signs. The representation is based on the interpretation of the lattice magnetization as the sum of the signed areas of clusters. If time permits, potential extensions, including to three dimensions, will also be discussed. The talk will be based on joint work with Chuck Newman and on work in progress with Chuck Newman and C. Garban.

Horng-Tzer Yau, Harvard University
Dyson's Sine Kernel, Wigner Random Matrices, and Interacting Particle Systems
Abstract:The local eigenvalue statistics of the Gaussian Unitary Ensemble (GUE) is given by Dyson's Sine kernel. It was conjectured that this law holds for a much general class of random matrices--- the universality conjecture of random matrices. For matrix ensembles that are unitarily invariant, there has been a great progress using technique from orthogonal polynomials. For the case of Hermitian Wigner random matrices i.e. for matrix ensembles with i.i.d. entries are in general not unitarily invariant, the only result is due to Johansson who proved the sine kernel for N by N matrices that are of the form $H + t V$ where $H$ is distributed according to a Wigner matrix ensemble and $V$ has the law of GUE. The parameter $t$ is required to be of order one. Our main result states that the Dyson's sine kernel holds for $t \ge N^{-3/4}$ i.e. for Wigner matrices with a vanishing Gaussian perturbation. Our approach is based on technique from interacting particle systems and key technical inputs are the local semi-circle law and level repulsion for Wigner random matrices. We remark that the universality conjecture for general Wigner matrices could be deduced from the case $t \ll N^{-1}$ which is still an open problem.

ClÈment Hongler, UniversitÈ de GenËve
The energy density in the 2D Ising model
Abstract:We study the Ising model from a conformal invariance point of view using discrete complex analysis methods. We are here interested in the scaling limit at critical temperature of the two-dimensional Ising model in a simply connected domain with boundary. In particular, we are interested in the effect of the boundary with some conditions (+ or free) on local observables. In this talk we will be interested in the behaviour of the so-called energy density field at the scaling limit, giving a rigourous exact derivation of predictions obtained using Conformal Field Theory, exhibiting a nice connection with hyperbolic geometry.
This derivation is made through the study of a so-called fermionic observable which is discrete holomorphic in a particular sense and converges to a holomorphic function in the scaling limit.
This is joint work with Stanislav Smirnov.

Past years
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