Abstract
We study long-range Bernoulli percolation on {mathbb {Z}}^d in which each two vertices x and y are connected by an edge with probability 1-exp (-beta Vert x-yVert ^{-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 Pβc(|K|≥n)≤Cn-(d-α)/(2d+α)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\mathbf {P}}_{\\beta _c}\\bigl (|K|\\ge n\\bigr ) \\le C n^{-(d-\\alpha )/(2d+\\alpha )} \\end{aligned}$$\\end{document}for every nge 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)le d(delta -1) relating the cluster-volume exponent delta and two-point function exponent eta .
Highlights
Let d ≥ 1 and suppose that J : Zd → [0, ∞) is both symmetric in the sense that J (x) = J (−x) for every x ∈ Zd and integrable in the sense that x∈Zd J (x) < ∞
For each β ≥ 0, long-range percolation on Zd with intensity J is the random graph with vertex set Zd in which we choose whether or not to include each potential edge {x, y} independently at random with inclusion probability 1−exp(−β J (y − x))
Studying the geometry of these clusters leads to many interesting questions, some of which are motivated by applications to modeling ‘small-world’ phenomena in physics, epidemiology, the social sciences, and so on; see e.g. [12, Section 1.4] and [14, Section 10.6] for background and many references
Summary
Remark 1.3 Theorem 1.2 leads to a new proof of a recent theorem of Xiang and Zou [76] which states that every countably infinite (but not necessarily finitely generated) group admits a symmetric, integrable function J : → [0, ∞) for which the associated weighted graph has a non-trivial percolation phase transition. To deduce their theorem from ours, pick a bijection σ : → {1, 2, . It would be interesting if a new proof of the results of [26] could be derived from Theorem 1.2 by comparison of short- and long-range percolation
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
