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Cover times of the massive random walk loop soup

Erik I. Broman, Federico Camia

Abstract

We study cover times of subsets of ${\mathbb Z}^2$ by a two-dimensional massive random walk loop soup. We consider a sequence of subsets $A_n \subset {\mathbb Z}^2$ such that $|A_n| \to \infty$ and determine the distributional limit of their cover times ${\mathcal T}(A_n).$ 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 $κ_n^{-1}=|A_n|^{1-8/(\log \log |A_n|)},$ showing that it can be described by a Gumbel distribution. Since a typical loop in this model will have length at most of order $κ_n^{-1/2}=|A_n|^{1/2},$ 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.

Cover times of the massive random walk loop soup

Abstract

We study cover times of subsets of by a two-dimensional massive random walk loop soup. We consider a sequence of subsets such that and determine the distributional limit of their cover times We allow the killing rate (or equivalently the ``mass'') of the loop soup to depend on the size of the set to be covered. In particular, we determine the limiting behavior of the cover times for inverse killing rates all the way up to showing that it can be described by a Gumbel distribution. Since a typical loop in this model will have length at most of order if exceeded the cover times of all points in a tightly packed set (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.
Paper Structure (7 sections, 20 theorems, 243 equations)

This paper contains 7 sections, 20 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. The loop soup
  3. Green's function estimates
  4. Probability estimates
  5. Second moment estimates
  6. Proof of main theorem
  7. Examples and discussion

Key Result

Theorem 1.1

Consider a sequence of finite subsets $A_n \subset {\mathbb{Z}}^2$ such that $|A_n| \uparrow \infty.$ Furthermore, assume that the killing rates $\kappa_n$ are such that, for every $n,$$\exp(e^{32})\leq \kappa^{-1}_n\leq |A_n|^{1-8/(\log \log |A_n|)}.$ We then have that for $n$ large enough and therefore where $G$ is a Gumbel distributed random variable.

Theorems & Definitions (43)

  • Theorem 1.1
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • Proposition 3.5
  • proof
  • Lemma 3.6
  • Proposition 3.7
  • ...and 33 more