Volume and heat kernel fluctuations for the three-dimensional uniform spanning tree

TL;DR

This paper investigates the fluctuations in volume and heat kernel of the 3D uniform spanning tree, revealing log-logarithmic variations around expected values, which enhances understanding of its geometric and probabilistic properties.

Contribution

It establishes the presence of log-logarithmic fluctuations in volume and heat kernel for the 3D uniform spanning tree, a novel insight into its stochastic structure.

Findings

Log-logarithmic fluctuations in volume of intrinsic balls

Similar fluctuations observed in the quenched heat kernel

Enhanced understanding of geometric and probabilistic behavior

Abstract

Let $\mathcal{U}$ be the uniform spanning tree on $\mathbb{Z}^{3}$. We show the occurrence of log-logarithmic fluctuations around the leading order for the volume of intrinsic balls in $\mathcal{U}$. As an application, we obtain similar fluctuations for the quenched heat kernel of the simple random walk on $\mathcal{U}$.