Second order term of cover time for planar simple random walk

TL;DR

This paper investigates the precise second order asymptotic term of the cover time for a simple random walk on a two-dimensional discrete torus, extending previous leading-term results.

Contribution

It provides the exact second order term for the cover time, advancing understanding of the precise asymptotics beyond the leading order.

Findings

Identifies the second order term in the cover time asymptotics.

Connects discrete random walk results with planar Brownian motion analysis.

Extends previous asymptotic results to exact second order terms.

Abstract

We consider the cover time for a simple random walk on the two-dimensional discrete torus of side length $n$. Dembo, Peres, Rosen, and Zeitouni [Ann. Math. 160:433-464, 2004] identified the leading term in the asymptotics for the cover time as $n$ goes to infinity. In this paper, we study the exact second order term. This is a discrete analogue of the work on the cover time for planar Brownian motion by Belius and Kistler [Probab. Theory Relat Fields. 167:461-552, 2017].