# Probability and Mathematical Physics Seminar

#### Scaling limits of the two- and three-dimensional uniform spanning trees and the associated random walks

**Speaker:**
David Croydon, Kyoto University

**Location:**
Warren Weaver Hall 1302

**Date:**
Friday, February 28, 2020, 11:10 a.m.

**Synopsis:**

I will describe recent work regarding the structure of and random walks on the two- and three-dimensional uniform spanning trees. The results from the two-dimensional case are from joint works with Martin Barlow (UBC) and Takashi Kumagai (Kyoto), and as well as a scaling limit, include demonstrating fluctuations in the on-diagonal part of the quenched heat kernel, and off-diagonal estimates for the averaged heat kernel. As for the three-dimensional situation, this is being investigated in an ongoing work with Omer Angel (UBC), Sarai Hernandez-Torres (UBC) and Daisuke Shiraishi (Kyoto). In the latter work, we demonstrate the tightness of the graph and the random walk's annealed law under rescaling, and convergence along a particular subsequence. We also derive the random walk's walk dimension (with respect to both the intrinsic and Euclidean metric) and its spectral dimension, as well as heat kernel estimates for any diffusion that arises as a scaling limit.