{"title":"Infinite collision property for the three-dimensional uniform spanning tree","authors":"Satomi Watanabe","doi":"10.1142/s2661335223500053","DOIUrl":null,"url":null,"abstract":"Let $\\mathcal{U}$ be the uniform spanning tree on $\\mathbb{Z}^3$, whose probability law is denoted by $\\mathbf{P}$. For $\\mathbf{P}$-a.s. realization of $\\mathcal{U}$, the recurrence of the the simple random walk on $\\mathcal{U}$ is proved in [5] and it is also demonstrated in [8] that two independent simple random walks on $\\mathcal{U}$ collide infinitely often. In this article, we will give a quantitative estimate on the number of collisions of two independent simple random walks on $\\mathcal{U}$, which provides another proof of the infinite collision property of $\\mathcal{U}$.","PeriodicalId":34218,"journal":{"name":"International Journal of Mathematics for Industry","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2023-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Mathematics for Industry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1142/s2661335223500053","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Let $\mathcal{U}$ be the uniform spanning tree on $\mathbb{Z}^3$, whose probability law is denoted by $\mathbf{P}$. For $\mathbf{P}$-a.s. realization of $\mathcal{U}$, the recurrence of the the simple random walk on $\mathcal{U}$ is proved in [5] and it is also demonstrated in [8] that two independent simple random walks on $\mathcal{U}$ collide infinitely often. In this article, we will give a quantitative estimate on the number of collisions of two independent simple random walks on $\mathcal{U}$, which provides another proof of the infinite collision property of $\mathcal{U}$.