Quenched and averaged tails of the heat kernel of the two-dimensional uniform spanning tree

TL;DR

This paper studies the heat kernel of the 2D uniform spanning tree, revealing log-logarithmic fluctuations, providing two-sided estimates, and analyzing scaling limits to refine asymptotic behaviors.

Contribution

It advances understanding of the heat kernel's fluctuations and scaling limits on the 2D uniform spanning tree, improving previous results with new estimates and asymptotics.

Findings

Log-logarithmic fluctuations in the quenched heat kernel

Two-sided estimates for the averaged heat kernel

Different exponents in quenched and averaged off-diagonal heat kernels

Abstract

This article investigates the heat kernel of the two-dimensional uniform spanning tree. We improve previous work by demonstrating the occurrence of log-logarithmic fluctuations around the leading order polynomial behaviour for the on-diagonal part of the quenched heat kernel. In addition we give two-sided estimates for the averaged heat kernel, and we show that the exponents that appear in the off-diagonal parts of the quenched and averaged versions of the heat kernel differ. Finally, we derive various scaling limits for the heat kernel, the implications of which include enabling us to sharpen the known asymptotics regarding the on-diagonal part of the averaged heat kernel and the expected distance travelled by the associated simple random walk.