{"title":"大规模随机漫步循环汤的覆盖时间","authors":"Erik I. Broman, Federico Camia","doi":"10.1007/s11040-024-09478-9","DOIUrl":null,"url":null,"abstract":"<div><p>We study cover times of subsets of <span>\\({\\mathbb {Z}}^2\\)</span> by a two-dimensional massive random walk loop soup. We consider a sequence of subsets <span>\\(A_n \\subset {\\mathbb {Z}}^2\\)</span> such that <span>\\(|A_n| \\rightarrow \\infty \\)</span> and determine the distributional limit of their cover times <span>\\({\\mathcal {T}}(A_n)\\)</span>. We allow the killing rate <span>\\(\\kappa _n\\)</span> (or equivalently the “mass”) of the loop soup to depend on the size of the set <span>\\(A_n\\)</span> to be covered. In particular, we determine the limiting behavior of the cover times for inverse killing rates all the way up to <span>\\(\\kappa _n^{-1}=|A_n|^{1-8/(\\log \\log |A_n|)},\\)</span> showing that it can be described by a Gumbel distribution. Since a typical loop in this model will have length at most of order <span>\\(\\kappa _n^{-1/2}=|A_n|^{1/2},\\)</span> if <span>\\(\\kappa _n^{-1}\\)</span> exceeded <span>\\(|A_n|,\\)</span> the cover times of all points in a tightly packed set <span>\\(A_n\\)</span> (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.\n</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"27 2","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-024-09478-9.pdf","citationCount":"0","resultStr":"{\"title\":\"Cover Times of the Massive Random Walk Loop Soup\",\"authors\":\"Erik I. Broman, Federico Camia\",\"doi\":\"10.1007/s11040-024-09478-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We study cover times of subsets of <span>\\\\({\\\\mathbb {Z}}^2\\\\)</span> by a two-dimensional massive random walk loop soup. We consider a sequence of subsets <span>\\\\(A_n \\\\subset {\\\\mathbb {Z}}^2\\\\)</span> such that <span>\\\\(|A_n| \\\\rightarrow \\\\infty \\\\)</span> and determine the distributional limit of their cover times <span>\\\\({\\\\mathcal {T}}(A_n)\\\\)</span>. We allow the killing rate <span>\\\\(\\\\kappa _n\\\\)</span> (or equivalently the “mass”) of the loop soup to depend on the size of the set <span>\\\\(A_n\\\\)</span> to be covered. In particular, we determine the limiting behavior of the cover times for inverse killing rates all the way up to <span>\\\\(\\\\kappa _n^{-1}=|A_n|^{1-8/(\\\\log \\\\log |A_n|)},\\\\)</span> showing that it can be described by a Gumbel distribution. Since a typical loop in this model will have length at most of order <span>\\\\(\\\\kappa _n^{-1/2}=|A_n|^{1/2},\\\\)</span> if <span>\\\\(\\\\kappa _n^{-1}\\\\)</span> exceeded <span>\\\\(|A_n|,\\\\)</span> the cover times of all points in a tightly packed set <span>\\\\(A_n\\\\)</span> (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.\\n</p></div>\",\"PeriodicalId\":694,\"journal\":{\"name\":\"Mathematical Physics, Analysis and Geometry\",\"volume\":\"27 2\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-03-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s11040-024-09478-9.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Physics, Analysis and Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s11040-024-09478-9\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Physics, Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s11040-024-09478-9","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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.
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.
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