Pointwise two-point function estimates and a non-perturbative proof of mean-field critical behaviour for long-range percolation

• 成果类型: Article; Early Access
• 署名作者: Hutchcroft, Tom
• 署名单位: California Institute of Technology
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01410-8
• 发表日期: 2025
• 关键词: critical exponents; bernoulli percolation; PHASE-TRANSITION; models; bounds; sharpness

摘要

In long-range percolation on Z(d), we connect each pair of distinct points x and y by an edge independently at random with probability 1-exp(-beta parallel to x-y parallel to(-d-alpha)), where alpha> 0 is fixed and beta >= 0 is a parameter. In a previous paper, we proved that if 0 < alpha < d then the critical two-point function satisfies the spatially averaged upper bound 1/ r(d) & sum;(d)( x is an element of[-r,r]) (P)beta(c)(0 <-> x) <= r(-d+alpha) for every r >= 1. This upper bound is believed to be sharp for values of alpha strictly below the crossover value alpha c(d), and a matching lower bound for alpha < 1 was proven by Baumler and Berger (AIHP 2022). In this paper, we prove pointwise upper and lower bounds of the same order under the same assumption that alpha < 1. We also prove analogous two-sided pointwise estimates on the slightly subcritical two-point function under the same hypotheses, interpolating between parallel to x parallel to(-d+alpha) decay below the correlation length and parallel to x parallel to(-d-alpha) decay above the correlation length. In dimensions d = 1, 2, 3, we deduce that the triangle condition holds under the minimal assumption that 0 < d/3. While this result had previously been established under additional perturbative assumptions using the lace expansion, our proof is completely non-perturbative and does not rely on the lace expansion in any way.