Cover times of the massive random walk loop soup

Abstract: We study cover times of subsets of ${\mathbb Z}<sup>2$ by a two-dimensional massive random walk loop soup. We consider a sequence of subsets $A_n \subset {\mathbb Z}<sup>2$ such that $|A_n| \to \infty$ 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 $\kappa_n<sup>{-1/2}=|A_n|<sup>{1/2},$ if $\kappa_n<sup>{-1}$ 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.