Cayley图的渗透阈值在1处的间隙

IF 1.5 Q2 PHYSICS, MATHEMATICAL Annales de l Institut Henri Poincare D Pub Date : 2021-10-31 DOI:10.1214/22-aihp1286

C. Panagiotis, Franco Severo

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引用次数: 3

摘要

我们证明了Cayley图的渗透阈值$p_c$的可能值集在1处有一个间隙，即存在$\varepsilon_0>0$，使得对于每个Cayley图$G$都有$p_c(G)=1$或$p_c(G) \leq 1-\varepsilon_0$。该证明建立在dumini - copin, Goswami, Raoufi, Severo＆Yadin利用高斯自由场证明相变存在性的新方法的基础上，结合Breuillard, Green＆Tao关于多项式生长群结构的Gromov定理的有限版本。

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Gap at 1 for the percolation threshold of Cayley graphs

We prove that the set of possible values for the percolation threshold $p_c$ of Cayley graphs has a gap at 1 in the sense that there exists $\varepsilon_0>0$ such that for every Cayley graph $G$ one either has $p_c(G)=1$ or $p_c(G) \leq 1-\varepsilon_0$. The proof builds on the new approach of Duminil-Copin, Goswami, Raoufi, Severo&Yadin to the existence of phase transition using the Gaussian free field, combined with the finitary version of Gromov's theorem on the structure of groups of polynomial growth of Breuillard, Green&Tao.

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