Cover Times of the Massive Random Walk Loop Soup

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2024年3月22日

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Erik I. Broman,
Federico Camia

摘要

We study cover times of subsets of ${\mathbb {Z}}^2$ $Z^{2}$ by a two-dimensional massive random walk loop soup. We consider a sequence of subsets $A_n \subset {\mathbb {Z}}^2$ $A_{n} \subset Z^{2}$ such that $|A_n| \rightarrow \infty $ $| A_{n} | \to \infty$ and determine the distributional limit of their cover times ${\mathcal {T}}(A_n)$ $T (A_{n})$ . We allow the killing rate $\kappa _n$ $κ_{n}$ (or equivalently the “mass”) of the loop soup to depend on the size of the set $A_n$ $A_{n}$ to be covered. In particular, we determine the limiting behavior of the cover times for inverse killing rates all the way up to $\kappa _n^{-1}=|A_n|^{1-8/(\log \log |A_n|)},$ $κ_{n}^{- 1} = {| A_{n} |}^{1 - 8 / (log log | A_{n} |)},$ showing that it can be described by a Gumbel distribution. Since a typical loop in this model will have length at most of order $\kappa _n^{-1/2}=|A_n|^{1/2},$ $κ_{n}^{- 1 / 2} = {| A_{n} |}^{1 / 2},$ if $\kappa _n^{-1}$ $κ_{n}^{- 1}$ exceeded $|A_n|,$ $| A_{n} |,$ the cover times of all points in a tightly packed set $A_n$ $A_{n}$ (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.