Nonconcentration of hitting times for random walks on graphs

TL;DR

This paper investigates the variability of hitting times in random walks on finite graphs, establishing bounds that depend on graph properties and disproving certain conjectures about their concentration.

Contribution

It provides new variance lower bounds for hitting times, constructs counterexamples to existing conjectures, and extends results to reversible Markov chains.

Findings

Variance of hitting times is bounded below by a function of the mean and graph size.

Constructed high-degree graphs with linear mean and bounded variance, disproving a conjecture.

Extended the main results to finite reversible Markov chains and showed a conjecture fails even for bounded-degree trees.

Abstract

We study nonconcentration of hitting times for simple random walk on finite graphs. We prove that, for every connected graph with $n$ vertices, \[ \operatorname{Var}_x(\tau_y)+\mathbb E_x\tau_y \ge \frac{(\mathbb E_x\tau_y)^2}{1+\log n}, \] with the logarithmic term sharp up to constants. Under a bounded-degree assumption the additive mean term can be removed, giving a variance lower bound depending only on \(\mathbb E_x\tau_y\) and the graph distance \(\dist(x,y)\). We show that this degree assumption is necessary by constructing high-degree graphs with linear mean and bounded variance; the same construction disproves a conjecture of Norris-Peres-Zhai concerning local nonconcentration of hitting times. We also prove a sharper tree estimate, extend the main argument to finite reversible Markov chains, and show that Holroyd's interval conjecture, stated in Norris-Peres-Zhai, fails even for bounded-degree trees.