{"title":"超树随机测量支持的全断开性和渗流","authors":"Edwin Perkins, Delphin Sénizergues","doi":"arxiv-2409.11841","DOIUrl":null,"url":null,"abstract":"Super-tree random measure's (STRM's) were introduced by Allouba, Durrett,\nHawkes and Perkins as a simple stochastic model which emulates a superprocess\nat a fixed time. A STRM $\\nu$ arises as the a.s. limit of a sequence of\nempirical measures for a discrete time particle system which undergoes\nindependent supercritical branching and independent random displacement\n(spatial motion) of children from their parents. We study the connectedness\nproperties of the closed support of a STRM ($\\mathrm{supp}(\\nu)$) for a\nparticular choice of random displacement. Our main results are distinct\nsufficient conditions for the a.s. total disconnectedness (TD) of\n$\\mathrm{supp}(\\nu)$, and for percolation on $\\mathrm{supp}(\\nu)$ which will\nimply a.s. existence of a non-trivial connected component in\n$\\mathrm{supp}(\\nu)$. We illustrate a close connection between a subclass of\nthese STRM's and super-Brownian motion (SBM). For these particular STRM's the\nabove results give a.s. TD of the support in three and higher dimensions and\nthe existence of a non-trivial connected component in two dimensions, with the\nthree-dimensional case being critical. The latter two-dimensional result\nassumes that $p_c(\\mathbb{Z}^2)$, the critical probability for site percolation\non $\\mathbb{Z}^2$, is less than $1-e^{-1}$. (There is strong numerical evidence\nsupporting this condition although the known rigorous bounds fall just short.)\nThis gives evidence that the same connectedness properties should hold for SBM.\nThe latter remains an interesting open problem in dimensions $2$ and $3$ ever\nsince it was first posed by Don Dawson over $30$ years ago.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"40 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Total disconnectedness and percolation for the supports of super-tree random measures\",\"authors\":\"Edwin Perkins, Delphin Sénizergues\",\"doi\":\"arxiv-2409.11841\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Super-tree random measure's (STRM's) were introduced by Allouba, Durrett,\\nHawkes and Perkins as a simple stochastic model which emulates a superprocess\\nat a fixed time. A STRM $\\\\nu$ arises as the a.s. limit of a sequence of\\nempirical measures for a discrete time particle system which undergoes\\nindependent supercritical branching and independent random displacement\\n(spatial motion) of children from their parents. We study the connectedness\\nproperties of the closed support of a STRM ($\\\\mathrm{supp}(\\\\nu)$) for a\\nparticular choice of random displacement. Our main results are distinct\\nsufficient conditions for the a.s. total disconnectedness (TD) of\\n$\\\\mathrm{supp}(\\\\nu)$, and for percolation on $\\\\mathrm{supp}(\\\\nu)$ which will\\nimply a.s. existence of a non-trivial connected component in\\n$\\\\mathrm{supp}(\\\\nu)$. We illustrate a close connection between a subclass of\\nthese STRM's and super-Brownian motion (SBM). For these particular STRM's the\\nabove results give a.s. TD of the support in three and higher dimensions and\\nthe existence of a non-trivial connected component in two dimensions, with the\\nthree-dimensional case being critical. The latter two-dimensional result\\nassumes that $p_c(\\\\mathbb{Z}^2)$, the critical probability for site percolation\\non $\\\\mathbb{Z}^2$, is less than $1-e^{-1}$. (There is strong numerical evidence\\nsupporting this condition although the known rigorous bounds fall just short.)\\nThis gives evidence that the same connectedness properties should hold for SBM.\\nThe latter remains an interesting open problem in dimensions $2$ and $3$ ever\\nsince it was first posed by Don Dawson over $30$ years ago.\",\"PeriodicalId\":501245,\"journal\":{\"name\":\"arXiv - MATH - Probability\",\"volume\":\"40 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Probability\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.11841\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11841","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Total disconnectedness and percolation for the supports of super-tree random measures
Super-tree random measure's (STRM's) were introduced by Allouba, Durrett,
Hawkes and Perkins as a simple stochastic model which emulates a superprocess
at a fixed time. A STRM $\nu$ arises as the a.s. limit of a sequence of
empirical measures for a discrete time particle system which undergoes
independent supercritical branching and independent random displacement
(spatial motion) of children from their parents. We study the connectedness
properties of the closed support of a STRM ($\mathrm{supp}(\nu)$) for a
particular choice of random displacement. Our main results are distinct
sufficient conditions for the a.s. total disconnectedness (TD) of
$\mathrm{supp}(\nu)$, and for percolation on $\mathrm{supp}(\nu)$ which will
imply a.s. existence of a non-trivial connected component in
$\mathrm{supp}(\nu)$. We illustrate a close connection between a subclass of
these STRM's and super-Brownian motion (SBM). For these particular STRM's the
above results give a.s. TD of the support in three and higher dimensions and
the existence of a non-trivial connected component in two dimensions, with the
three-dimensional case being critical. The latter two-dimensional result
assumes that $p_c(\mathbb{Z}^2)$, the critical probability for site percolation
on $\mathbb{Z}^2$, is less than $1-e^{-1}$. (There is strong numerical evidence
supporting this condition although the known rigorous bounds fall just short.)
This gives evidence that the same connectedness properties should hold for SBM.
The latter remains an interesting open problem in dimensions $2$ and $3$ ever
since it was first posed by Don Dawson over $30$ years ago.