Probability and Mathematical Physics Seminar

Synopsis:

The eigenvalues of a uniformly distributed unitary matrix (CUE) have the physical interpretation of a system of particles subject to a logarithmic pair interaction, restricted to lie on the unit circle and at inverse temperature 2. In this talk, I will present a more general model in which the unit circle is replaced by a sufficiently regular Jordan curve. In a paper with Johansson, we obtained the asymptotic partition function and the Laplace transform of linear statistics at any positive temperature. These can be expressed using either the exterior conformal mapping of the curve or its associated Grunsky operator.

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Liouville quantum gravity (LQG) is a natural model describing random surfaces. A powerful tool of studying LQG is the conformal welding, where we glue two LQG surfaces together into a single LQG surface with the interface being Schramm-Loewner evolution (SLE) type curves. We will present some recent results on the conformal welding of LQG surfaces. We will also discuss their extensions and applications, including connections with reversibility of SLE curves, finiteness of multiple SLE partition functions and conformal loop ensemble (CLE) boundary touching probability. Based on joint works with Ang, Holden, Sun and Zhuang.

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The Probability and the City seminar is a joint meeting of the probability seminars at Courant and at Columbia, held twice every semester (hosted once at each institution).
The second meeting of Fall 2023 will feature:

• Amol Aggarwal (Columbia University) 
  
  Title:
  
  Abstract:

• Allan Sly (Princeton University)
  
  Title: Rotationally invariant first passage percolation: Concentration and scaling relations
  
  Abstract: For rotationally invariant first passage percolation on the plane, we use a multi-scale argument to prove stretched exponential concentration of the passage times at the scale of the standard deviation. Our results are proved for several standard rotationally invariant models of first passage percolation, e.g. Riemannian FPP, Voronoi FPP and the Howard-Newman model. As a consequence, we prove a version of the scaling relations between the passage times fluctuation and transversal fluctuations of geodesics.  These are the first such unconditional results.
   

Notes:

 

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Synopsis:

What is the behavior of the longest increasing subsequence of a uniformly random permutation? Its length is of order 2n^{1/2} plus Tracy–Widom fluctuations of order n^{1/6}. Its scaling limit is the directed geodesic of the directed landscape.

This talk discusses how this behavior changes dramatically when one looks at universal Brownian–type permutations, i.e., permutations sampled from the Brownian separable permutons. We show that there are explicit constants 1/2<α<β<1 such that the length of the longest increasing subsequence in a random permutation of size n sampled from the Brownian separable permutons is between n^{α–o(1)} and n^{β+o(1)} with high probability. We present numerical simulations which suggest that the lower bound is close to optimal. Our proofs are based on the analysis of a fragmentation process embedded in a Brownian excursion introduced by Bertoin (2002).

If time permits, we conclude by discussing some conjectures for permutations sampled from the skew Brownian permutons, a model of universal permutons generalizing the Brownian separable permutons: here, the longest increasing subsequences should be closely related with some models of random directed metrics on planar maps.

Based on joint work with William Da Silva and Ewain Gwynne.

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Synopsis:

The discovery by McCann of displacement convexity had a significant impact on probability, analysis, and geometry. I will introduce a new and stronger notion of displacement convexity which operates on the matrix level. I will then show that a broad class of flows satisfy matrix displacement convexity: heat flow, optimal transport, entropic interpolation, mean-field games, and semiclassical limits of non-linear Schrödinger equations. Consequently, the ambient dimensions of functional inequalities describing the behavior of these flows can be replaced by their intrinsic dimensions, capturing the behavior of the  flows along different directions in space. This leads to intrinsic dimensional functional inequalities which provide a systematic improvement on numerous classical functional inequalities.

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The Airy line ensemble was introduced by Prähofer and Spohn and is conjectured to describe the scaling limit of various random surfaces and stochastic growth models in the Kardar–Parisi–Zhang universality class. In this talk I will discuss a characterization result for Airy line ensembles, essentially indicating that if the top curve of a Brownian line ensemble is within a multiplicative 1+o(1) from a parabola, then it must be the Airy line ensemble (up to an affine shift). This is a joint work with Amol Aggarwal.

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Synopsis:

Why some stationary planar point processes generate stationary fields with divergence equaled the counting measure of the point process minus the Lebesgue measure (the infinite Ginibre determinantal process and the zero set of the Gaussian Entire Function belong to this class), while others (like the Poisson point process) don't? In the talk we give a simple answer to this question and discuss curious properties of that stationary field, for instance, the logarithmic divergence of their covariance, and the fluctuations of their line integrals. 

A central role will be played by random meromorphic functions having simple poles with unit residues at a given stationary point process. These random meromorphic functions can be viewed as random analogues of the Weierstrass zeta function from the theory of elliptic functions.

If time permits, we will also touch existence of somewhat counter-intuitive  exotic objects generated by stationary planar point processes.

The talk will be based on the joint work with Oren Yakir and Aron Wennman: 

https://arxiv.org/abs/2210.09882 , https://arxiv.org/abs/2211.01312

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Synopsis:

The 2D XY model has attracted attention of physicists and mathematicians for several decades. One way to understand this model is through its dual height function. Recent developments make it possible to show that the phase transitions of the two models coincide. The talk will highlight several connections between the two models and is based on arXiv:2301.06905 (Bijecting the BKT transition) and arXiv:2211.14365 (A dichotomy theory for height functions).

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The Fisher-KPP equation was introduced in 1937 to model the spread of an advantageous gene through a spatially distributed population. Remarkably precise information on the traveling front has been obtained via a connection with branching Brownian motion, beginning with works of McKean and Bramson in the 70s. I will discuss an extension of this probabilistic approach to the Road-Field Model: a reaction-diffusion PDE system introduced by H. Berestycki et al. to describe enhancement of biological invasions by a line of fast diffusion, such as a river or a road. Based on joint work with Amir Dembo.

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Quasisymmetric maps are a metric space analogue of conformal maps. The conformal dimension of a metric space is the infimum of the Hausdorff dimension among all metric spaces that are quasisymmetric to the given space.  Conformal dimension was introduced by Pansu (1989) to study boundaries of Gromov hyperbolic spaces. More recently, the notion of conformal walk dimension was introduced to study Harnack inequalities for symmetric diffusion processes. An important application of conformal walk dimension is a generalization of Moser's stability result for Harnack inequality in the Euclidean space to manifolds and more generally spaces equipped with symmetric diffusions.  This talk will survey aspects of conformal dimensions in analysis and probability highlighting some parallels and differences.

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Synopsis:

The Probability and the City seminar is a joint meeting of the probability seminars at Courant and at Columbia, held twice every semester (hosted once at each institution).
The first meeting of Fall 2023 will feature:

• Simona Diaconu (Courant)
  Title: Traces, Eigenvalues, and Eigenvectors Abstract

• Abstract: 
  Random matrix theory is concerned primarily with the eigenvalues and eigenvectors of certain ensembles. The method of moments is a common technique in this area: it goes back to Wigner in 1955, and consists of finding the asymptotic behavior of traces of large powers of matrices. The most common uses are operator norm bounds with high probability, a classical result of Soshnikov from 1998 being edge eigenspectrum universality. The purpose of this talk is highlighting the versatility of this technique by presenting three (more) recent applications, two yielding central limit theorems for eigenvalues, and one perhaps even surprising, related to eigenvectors. No background in random matrix theory is assumed.

• Yu Gu (University of Maryland)
  Title: Effective diffusivities in periodic KPZ Abstract
  
  Abstract: 
  It’s known that the KPZ equation on a torus exhibits diffusive behaviors in large time. In this talk, I will discuss the recent joint work with Tomasz Komorowski and Alex Dunlap in which we derived nearly explicit expressions of the diffusion constants. The argument is based on a combination of Malliavin calculus and stochastic homogenization.

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Quantum ergodicity (QE) is a notion of eigenvector delocalization, that large eigenvector entries are “well spread” throughout the entire graph. Such a property is true of ``chaotic’’ manifolds and graphs, such as random regular graphs and Riemannian manifolds with ergodic geodesic flow. Focusing on graphs, outside of very specific examples, QE was previously only known to hold for families of graphs with a tree local limit. In this talk we show how QE is in satisfied for many families of operators on periodic graphs, including Schrodinger operators with periodic potential on the discrete torus and on the honeycomb lattice.

In order to do this, we use new ideas coming from analyzing Bloch varieties, and some methods coming from proofs in the continuous setting.

Based on joint work with Mostafa Sabri.

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Synopsis:

The broad aim of the talk is to describe the dynamics of interacting particle systems on rare dynamical events. For many models large deviation results are available and give an estimate of the probability of such a rare event. The point here is to describe the trajectories belonging to this rare event, in the high-dimensional limit where the number of particles is large.

From a technical point of view, the difficulty is that the original dynamics "conditioned" (in a suitable sense) to the rare dynamical event is high-dimensional, in general non-local (even if the original dynamics were local), non homogeneous in time and not explicitly known. The study of such dynamics on rare events is of particular interest when the interacting particle system undergoes a dynamical phase transition.

I will consider a simple yet already very rich model: the Weakly Asymmetric Simple Exclusion Process (WASEP) on a ring, subject to an atypical current of particles. The WASEP has a dynamical phase transition when the current strength exceeds a certain threshold. Using so-called relative entropy techniques, it is possible to obtain a detailed description of the WASEP dynamics tilted by the current. In particular I can compute spatial (two-point) correlations, confirming 2007 heuristic computations of Thierry Bodineau and Bernard Derrida and giving information on the dynamical phase transition.

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