Giant component for the supercritical level-set percolation of the Gaussian free field on regular expander graphs

IF 3.1 1区数学 Q1 MATHEMATICS Communications on Pure and Applied Mathematics Pub Date : 2023-06-28 DOI:10.1002/cpa.22112

Jiří Černý

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引用次数: 1

Abstract

We consider the zero-average Gaussian free field on a certain class of finite d-regular graphs for fixed $d \geq 3$ . 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 $h < h_{★}$ , 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 $h < h_{★}$ , 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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正则膨胀图上高斯自由场超临界水平集渗流的巨分量

我们考虑一类固定d≥3$d\ge3$的有限d正则图上的零平均高斯自由场。此类包括大周长的d-正则扩展器和随机d-正则图的典型实现。我们证明了在h能级以上的零平均高斯自由场的能级集在整个超临界相中有一个巨大的分量，即对于所有h

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来源期刊

Communications on Pure and Applied Mathematics 数学-数学

CiteScore

6.70

自引率

3.30%

发文量

审稿时长

>12 weeks

期刊介绍： Communications on Pure and Applied Mathematics (ISSN 0010-3640) is published monthly, one volume per year, by John Wiley & Sons, Inc. © 2019. The journal primarily publishes papers originating at or solicited by the Courant Institute of Mathematical Sciences. It features recent developments in applied mathematics, mathematical physics, and mathematical analysis. The topics include partial differential equations, computer science, and applied mathematics. CPAM is devoted to mathematical contributions to the sciences; both theoretical and applied papers, of original or expository type, are included.