THE CRITICAL TWO-POINT FUNCTION FOR LONG-RANGE PERCOLATION ON THE HIERARCHICAL LATTICE

• 成果类型: Article
• 署名作者: Hutchcroft, Tom
• 署名单位: California Institute of Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/23-AAP1982
• 发表日期: 2024
• 页码: 986-1002
• 关键词: self-avoiding walk; critical-behavior; PHASE-TRANSITION; greens-function; ising-model; transience; recurrence; sharpness; exponents; points

摘要

We prove up-to-constants bounds on the two-point function (i.e., point-to-point connection probabilities) for critical long-range percolation on the d-dimensional hierarchical lattice. More precisely, we prove that if we connect each pair of points x and y by an edge with probability 1 - exp(- beta parallel to x - y parallel to(-d-alpha)), where 0< alpha < d is fixed and beta >= 0 is a parameter, then the critical two-point function satisfies P-beta c (x <-> y) (sic) parallel to x - y parallel to(-d+alpha) for every pair of distinct points x and y. We deduce in particular that the model has mean-field critical behaviour when alpha < d/3 and does not have mean-field critical behaviour when alpha > d/3.