Cover Times of the Massive Random Walk Loop Soup

We study cover times of subsets of Z² by a two-dimensional massive random walk loop soup. We consider a sequence of subsets A_n⊂Z² such that |A_n|→∞ and determine the distributional limit of their cover times T(A_n). We allow the killing rate κ_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 κ_n^-1=|An|^{1-8/(loglog|An|)}, showing that it can be described by a Gumbel distribution. Since a typical loop in this model will have length at most of order κ_n^-1/2=|An|^1/2, if κ_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.