Cover Times of the Massive Random Walk Loop Soup

Abstract We study cover times of subsets of {Z}² Z 2 by a two-dimensional massive random walk loop soup. We consider a sequence of subsets Aₙ {Z}² A n ⊂ Z 2 such that |Aₙ| | A n | → ∞ and determine the distributional limit of their cover times {T} (Aₙ) T (A n). We allow the killing rate ₙ κ n (or equivalently the “mass”) of the loop soup to depend on the size of the set Aₙ 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 ₙ^-1=|Aₙ|^1-8/ (|Aₙ|), κ 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 ₙ^-1/2=|Aₙ|^1/2, κ n - 1 / 2 = | A n | 1 / 2, if ₙ^-1 κ n - 1 exceeded |Aₙ|, | A n |, the cover times of all points in a tightly packed set Aₙ 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.