Cover Times of the Massive Random Walk Loop Soup

Journal article, 2024

We study cover times of subsets of Z2 by a two-dimensional massive random walk loop soup. We consider a sequence of subsets An⊂Z2 such that |An|→∞ and determine the distributional limit of their cover times T(An). We allow the killing rate κn (or equivalently the “mass”) of the loop soup to depend on the size of the set An to be covered. In particular, we determine the limiting behavior of the cover times for inverse killing rates all the way up to κn-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 κn-1/2=|An|1/2, if κn-1 exceeded |An|, the cover times of all points in a tightly packed set An (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.