Power-law bounds for critical long-range percolation below the upper-critical dimension

• 成果类型: Article
• 署名作者: Hutchcroft, Tom
• 署名单位: University of Cambridge
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-021-01043-7
• 发表日期: 2021
• 页码: 533-570
• 关键词: critical 2-point functions; mean-field behavior; critical exponents; infinite cluster; bernoulli percolation; chemical distance; PHASE-TRANSITION; lace expansion; inequalities; models

摘要

We study long-range Bernoulli percolation on Z(d) in which each two vertices x and y are connected by an edge with probability 1 - exp(-beta parallel to x - y parallel to(-d-alpha)). It is a theorem of Noam Berger (Commun. Math. Phys., 2002) that if 0 < alpha < d then there is no infinite cluster at the critical parameter beta(c). We give a new, quantitative proof of this theorem establishing the power-law upper bound P-beta c (vertical bar K vertical bar >= n) <= Cn(-(d-alpha)/(2d+alpha)) for every n >= 1, where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality (2 - eta)(delta + 1) <= d(delta -1) relating the cluster-volume exponent delta and two-point function exponent eta.