Cover Times of the Massive Random Walk Loop Soup

IF 0.9 3区数学 Q3 MATHEMATICS, APPLIED

Mathematical Physics, Analysis and Geometry Pub Date : 2024-03-22 DOI:10.1007/s11040-024-09478-9

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引用次数: 0

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 \(|A_n| \rightarrow \infty \) and determine the distributional limit of their cover times \({\mathcal {T}}(A_n)\) . We allow the killing rate \(\kappa _n\) (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 \(\kappa _n^{-1}=|A_n|^{1-8/(\log \log |A_n|)},\) showing that it can be described by a Gumbel distribution. Since a typical loop in this model will have length at most of order \(\kappa _n^{-1/2}=|A_n|^{1/2},\) if \(\kappa _n^{-1}\) 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.

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大规模随机漫步循环汤的覆盖时间

摘要我们研究二维大规模随机游走环汤对\({\mathbb {Z}}^2\) 子集的覆盖时间。我们考虑一系列子集 \(A_n \子集 {\mathbb {Z}}^2\) 如 \(|A_n| \rightarrow \infty \)，并确定它们的覆盖时间的分布极限 \({\mathcal {T}}(A_n)\) 。我们允许环汤的杀灭率（或等同于 "质量"）取决于要覆盖的集合的大小（\(A_n\)）。特别是，我们确定了反向杀伤率一直到 \(\kappa _n^{-1}=|A_n|^{1-8/(\log \log |A_n|)},\)的覆盖时间的极限行为，表明它可以用甘贝尔分布来描述。如果 \(\kappa _n^{-1}\)超过 \(|A_n|,\)，那么这个模型中典型的环的长度最多为 \(\kappa_n^{-1/2}=|A_n|^{1/2},\)阶。我们的结果接近于这种极端情况。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Physics, Analysis and Geometry 数学-物理：数学物理

CiteScore

2.10

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.