Cover Times of the Massive Random Walk Loop Soup

Abstract

We study cover times of subsets of \({\mathbb {Z}}^2\) by a two-dimensional massive random walk loop soup. We consider a sequence of subsets \(A_n \subset {\mathbb {Z}}^2\) such that \(|A_n| \rightarrow \infty \) and determine the distributional limit of their cover times \({\mathcal {T}}(A_n)\). We allow the killing rate \(\kappa _n\) (or equivalently the “mass”) of the loop soup to depend on the size of the set \(A_n\) to be covered. In particular, we determine the limiting behavior of the cover times for inverse killing rates all the way up to \(\kappa _n^{-1}=|A_n|^{1-8/(\log \log |A_n|)},\) showing that it can be described by a Gumbel distribution. Since a typical loop in this model will have length at most of order \(\kappa _n^{-1/2}=|A_n|^{1/2},\) if \(\kappa _n^{-1}\) exceeded \(|A_n|,\) the cover times of all points in a tightly packed set \(A_n\) (i.e., a square or close to a ball) would presumably be heavily correlated, complicating the analysis. Our result comes close to this extreme case.