{"title":"正则膨胀图上高斯自由场超临界水平集渗流的巨分量","authors":"Jiří Černý","doi":"10.1002/cpa.22112","DOIUrl":null,"url":null,"abstract":"<p>We consider the zero-average Gaussian free field on a certain class of finite <i>d</i>-regular graphs for fixed <math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>≥</mo>\n <mn>3</mn>\n </mrow>\n <annotation>$d\\ge 3$</annotation>\n </semantics></math>. This class includes <i>d</i>-regular expanders of large girth and typical realisations of random <i>d</i>-regular graphs. We show that the level set of the zero-average Gaussian free field above level <i>h</i> has a giant component in the whole supercritical phase, that is for all <math>\n <semantics>\n <mrow>\n <mi>h</mi>\n <mo><</mo>\n <msub>\n <mi>h</mi>\n <mi>★</mi>\n </msub>\n </mrow>\n <annotation>$h<h_\\star$</annotation>\n </semantics></math>, with probability tending to one as the size of the graphs tends to infinity. In addition, we show that this component is unique. This significantly improves the result of [4], where it was shown that a linear fraction of vertices is in mesoscopic components if <math>\n <semantics>\n <mrow>\n <mi>h</mi>\n <mo><</mo>\n <msub>\n <mi>h</mi>\n <mi>★</mi>\n </msub>\n </mrow>\n <annotation>$h<h_\\star$</annotation>\n </semantics></math>, and together with the description of the subcritical phase from [4] establishes a fully-fledged percolation phase transition for the model.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.1000,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Giant component for the supercritical level-set percolation of the Gaussian free field on regular expander graphs\",\"authors\":\"Jiří Černý\",\"doi\":\"10.1002/cpa.22112\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We consider the zero-average Gaussian free field on a certain class of finite <i>d</i>-regular graphs for fixed <math>\\n <semantics>\\n <mrow>\\n <mi>d</mi>\\n <mo>≥</mo>\\n <mn>3</mn>\\n </mrow>\\n <annotation>$d\\\\ge 3$</annotation>\\n </semantics></math>. This class includes <i>d</i>-regular expanders of large girth and typical realisations of random <i>d</i>-regular graphs. We show that the level set of the zero-average Gaussian free field above level <i>h</i> has a giant component in the whole supercritical phase, that is for all <math>\\n <semantics>\\n <mrow>\\n <mi>h</mi>\\n <mo><</mo>\\n <msub>\\n <mi>h</mi>\\n <mi>★</mi>\\n </msub>\\n </mrow>\\n <annotation>$h<h_\\\\star$</annotation>\\n </semantics></math>, with probability tending to one as the size of the graphs tends to infinity. In addition, we show that this component is unique. This significantly improves the result of [4], where it was shown that a linear fraction of vertices is in mesoscopic components if <math>\\n <semantics>\\n <mrow>\\n <mi>h</mi>\\n <mo><</mo>\\n <msub>\\n <mi>h</mi>\\n <mi>★</mi>\\n </msub>\\n </mrow>\\n <annotation>$h<h_\\\\star$</annotation>\\n </semantics></math>, and together with the description of the subcritical phase from [4] establishes a fully-fledged percolation phase transition for the model.</p>\",\"PeriodicalId\":10601,\"journal\":{\"name\":\"Communications on Pure and Applied Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":3.1000,\"publicationDate\":\"2023-06-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications on Pure and Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22112\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications on Pure and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22112","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Giant component for the supercritical level-set percolation of the Gaussian free field on regular expander graphs
We consider the zero-average Gaussian free field on a certain class of finite d-regular graphs for fixed . This class includes d-regular expanders of large girth and typical realisations of random d-regular graphs. We show that the level set of the zero-average Gaussian free field above level h has a giant component in the whole supercritical phase, that is for all , with probability tending to one as the size of the graphs tends to infinity. In addition, we show that this component is unique. This significantly improves the result of [4], where it was shown that a linear fraction of vertices is in mesoscopic components if , and together with the description of the subcritical phase from [4] establishes a fully-fledged percolation phase transition for the model.
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