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

TL;DR

This paper provides a new quantitative power-law upper bound for the size of clusters at criticality in long-range percolation models below the upper-critical dimension, advancing understanding of non-mean-field critical behavior.

Contribution

The authors establish the first rigorous power-law upper bound for non-planar, non-mean-field Bernoulli percolation models, and introduce a universal inequality linking cluster exponents.

Findings

Proved a power-law upper bound for cluster size distribution at criticality.

Established a universal inequality relating cluster-volume and two-point function exponents.

Demonstrated the bound applies to models beyond planar or mean-field cases.

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 $1-\exp(-\beta \|x-y\|^{-d-\alpha})$. It is a theorem of Noam Berger (CMP, 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 \[ \mathbf{P}_{\beta_c}\bigl(|K|\geq n\bigr) \leq C n^{-(d-\alpha)/(2d+\alpha)} \] for every $n\geq 1$, 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 $(2-\eta)(\delta+1)\leq d(\delta-1)$ relating the cluster-volume exponent $\delta$ and two-point function exponent $\eta$.