Analyticity Results in Bernoulli Percolation

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS ACS Applied Bio Materials Pub Date : 2018-09-10 DOI:10.1090/memo/1431

Agelos Georgakopoulos, C. Panagiotis

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引用次数: 13

Abstract

We prove that for Bernoulli percolation on Z d \mathbb {Z}^d , d ≥ 2 d\geq 2 , the percolation density is an analytic function of the parameter in the supercritical interval. For this we introduce some techniques that have further implications. In particular, we prove that the susceptibility is analytic in the subcritical interval for all transitive short- or long-range models, and that p c b o n d > 1 / 2 p_c^{bond} >1/2 for certain families of triangulations for which Benjamini & Schramm conjectured that p c s i t e ≤ 1 / 2 p_c^{site} \leq 1/2 .

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伯努利渗流的分析结果

证明了在Z d \mathbb Z{^d, d≥2 d }\geq 2上的伯努利渗流，渗流密度是超临界区间参数的解析函数。为此，我们将介绍一些具有进一步含义的技术。特别地，我们证明了对于所有传递的短期或长期模型，在亚临界区间的磁化率是解析的。对于某些三角划分族，Benjamini ＆ Schramm推测p c site≤1/2 p＿c^{site}{}\leq 1/2, p c bo on和>1/2 p＿c^bond >1/2。

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