Wednesday Nov. 13, 4 pm (Probability and Analysis Seminar)
Room 6627

Title: The convergence of two dimensional loop-erased random walk to SLE(2)

Abstract:
In the colloquium talk on Thursday we will give an overview explaining the SLE processes and how they appear as scaling limits of various statistical physics models in two dimensions. The Wednesday talk will attempt to describe the proof of one instance of this principle: Take a simple random walk on a fine grid in a planar domain D, stop when you hit the boundary of D, and erase loops from the path as they are created. As the mesh refines, the limit of this process is the path of SLE(2). This result is joint work with Grag Lawler and Wendelin Werner.

Thursday Nov. 14, 4 pm (Colloquium)
Room 6627

Title: Emergence of symmetry: conformal invariance of scaling limits of random systems

Abstract:
A simple random walk on the square grid in the plane, when appropriately scaled, converges to Brownian motion. Brownian motion enjoys rotational symmetry and even conformal invariance, while the simple random walk does not. Recently, it was shown by several authors that other natural random two-dimensional systems (such as critical percolation, loop-erased random walks and uniform spanning trees) have conformally invariant scaling limits, and the limits have been explicitly described. In this talk, we will describe the random systems and their limits, discuss the implications of these results, and present some open problems.