Asymptotics of cover times via Gaussian free fields: Bounded-degree graphs and general trees

TL;DR

This paper establishes a new asymptotic relation between cover times of random walks and Gaussian free fields on bounded-degree graphs and trees, extending previous results beyond regular structures.

Contribution

It generalizes the asymptotic formula for cover times to broader classes of graphs and trees, and proves exponential concentration for cover times on general trees.

Findings

Cover time asymptotics relate to Gaussian free field supremum.

Asymptotic equality holds for bounded degree graphs and trees.

Exponential concentration bounds for cover times on trees.

Abstract

In this paper we show that on bounded degree graphs and general trees, the cover time of the simple random walk is asymptotically equal to the product of the number of edges and the square of the expected supremum of the Gaussian free field on the graph, assuming that the maximal hitting time is significantly smaller than the cover time. Previously, this was only proved for regular trees and the 2D lattice. Furthermore, for general trees, we derive exponential concentration for the cover time, which implies that the standard deviation of the cover time is bounded by the geometric mean of the cover time and the maximal hitting time.