On cover times for 2D lattices

TL;DR

This paper analyzes the cover time of random walks on 2D lattices, providing precise asymptotic estimates that improve previous results and support existing conjectures.

Contribution

It offers refined asymptotic bounds for the cover time on 2D lattices, advancing understanding of random walk behavior on these structures.

Findings

Asymptotic formula for cover time on 2D lattices

Improved bounds over previous results

Progress towards conjectures by Bramson and Zeitouni

Abstract

We study the cover time $\tau_{\mathrm{cov}}$ by (continuous-time) random walk on the 2D box of side length $n$ with wired boundary or on the 2D torus, and show that in both cases with probability approaching 1 as $n$ increases, $\sqrt{\tau_{\mathrm{cov}}}=\sqrt{2n^2}[\sqrt{2/\pi} \log n + O(\log\log n)]$. This improves a result of Dembo, Peres, Rosen, and Zeitouni (2004) and makes progress towards a conjecture of Bramson and Zeitouni (2009).