Covering of an inner subset by the confined random walk

TL;DR

This paper studies the cover time of inner subsets by a confined random walk in high-dimensional domains, revealing asymptotic behaviors involving Gumbel distributions and Poisson point processes, extending previous results to more general settings.

Contribution

It introduces a new analysis of the cover time and late points for confined random walks, including asymptotic expansions and distributional limits, generalizing prior work on simpler models.

Findings

Asymptotic expansion of cover time involving Gumbel distribution.

Poisson distribution of late points in the subset.

Extension of known results to more general 'ball-like' subsets.

Abstract

We consider the simple random walk conditioned to stay forever in a finite domain $D_N \subset \mathbb{Z}^d, d \geq 3$ of typical size $N$. This confined walk is a random walk on the conductances given by the first eigenvector of the Laplacian on $D_N$. On inner sets of $D_N$, the trace of this confined walk can be approximated by tilted random interlacements, which is a useful tool to understand some properties of the walk. In this paper, we propose to study the cover time of inner subsets $\Lambda_N$ of $D_N$ as well as the so-called late points of these subsets. If $\Lambda_N$ contains enough late points, we obtain the asymptotic expansion of the covering time as $c_\Lambda N^d \big[ \log N - \log\log N + \mathcal{G} \big]$, with $\mathcal{G}$ a Gumbel random variable, as well as a Poisson repartition of these late points. The method we use is similar to Belius' work about the simple random walk on the torus, which displays the same asymptotics albeit without the $\log \log N$ term. In the more general setting of ``ball-like'' $\Lambda_N$, we simply get the first term of the asymptotic expansion.