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

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 $\mathcal{U}$.