Tightness for the Cover Time of Wired Planar Domains

TL;DR

This paper analyzes the cover time of a wired planar domain random walk, establishing tight bounds and confirming a conjecture by comparing with the discrete Gaussian free field.

Contribution

It proves a tightness result for the cover time of wired planar domains, confirming a conjecture and linking it to the extremal landscape of the discrete Gaussian free field.

Findings

Square root of cover time normalized by size is tight.

Asymptotic behavior matches a conjecture by Bramson and Zeitouni.

Comparison with the discrete Gaussian free field is key to the proof.

Abstract

We consider a continuous time simple random walk on a subset of the square lattice with wired boundary conditions: the walk transitions at unit edge rate on the graph obtained from the lattice closure of the subset by contracting the boundary into one vertex. We study the cover time of such walk, namely the time it takes for the walk to visit all vertices in the graph. Taking a sequence of subsets obtained as scaled lattice versions of a nice planar domain, we show that the square root of the cover time normalized by the size of the subset, is tight around $\frac{1}{\sqrt{\pi}} \log N - \frac{1}{4 \sqrt{\pi}} \log \log N$, where $N$ is the scale parameter. This proves an analog, for the wired case, of a conjecture by Bramson and Zeitouni from 2009. The proof is based on comparison with the extremal landscape of the discrete Gaussian free field.