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 Aₙ Z² such that |Aₙ| and determine the distributional limit of their cover times T (Aₙ). We allow the killing rate ₙ (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 ₙ^-1=|Aₙ|^1-8/ (|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 ₙ^-1/2=|Aₙ|^1/2, if ₙ^-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.