## Abstract

We study cover times of subsets of Z^{2} by a two-dimensional massive random walk loop soup. We consider a sequence of subsets A_{n}⊂Z^{2} 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.

Original language | English (US) |
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Article number | 6 |

Journal | Mathematical Physics Analysis and Geometry |

Volume | 27 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2024 |

## Keywords

- 60G50
- 60K35
- Cover times
- Killing rates
- Random walk loop soup

## ASJC Scopus subject areas

- Mathematical Physics
- Geometry and Topology