# Student Probability Seminar

#### Critical and Near-Critical Percolation in Two Dimensions (And a Little Bit of SLE(6))

**Speaker:**
Pierre Nolin

**Location:**
Warren Weaver Hall 202

**Date:**
Friday, October 16, 2009, 3 p.m.

**Synopsis:**

Percolation is probably one of the simplest models of a random medium, but it nonetheless features the typical properties of most statistical mechanics systems, such as the existence of a phase transition and of critical exponents. After a general overview of this model, we will present techniques which were developed to describe two-dimensional percolation near its phase transition (mostly on the triangular lattice). This uses Smirnov's proof of conformal invariance in the scaling limit at criticality, results on the so-called Schramm-Loewner Evolution (SLE) obtained by Lawler, Schramm and Werner, and also Kesten's scaling relations that relate the near-critical regime to the critical regime itself.