On the derivation of mean-field percolation critical exponents from the triangle condition

TL;DR

This paper presents a new, quantitatively improved derivation of mean-field percolation critical exponents from the triangle condition, especially effective when the triangle diagram diverges slowly, with applications to long-range percolation.

Contribution

It introduces a novel method for deriving mean-field critical behaviour from the triangle condition, effective even when the triangle diagram diverges slowly, and applies this to hierarchical lattices.

Findings

Mean-field critical behaviour holds with polylogarithmic factors when the triangle diagram diverges slowly.

The new method improves bounds for the triangle diagram in percolation models.

Application to long-range percolation on hierarchical lattices confirms mean-field behaviour at the upper-critical dimension.

Abstract

We give a new derivation of mean-field percolation critical behaviour from the triangle condition that is quantitatively much better than previous proofs when the triangle diagram $\nabla_{p_c}$ is large. In contrast to earlier methods, our approach continues to yield bounds of reasonable order when the triangle diagram $\nabla_p$ is unbounded but diverges slowly as $p \uparrow p_c$, as is expected to occur in percolation on $\mathbb{Z}^d$ at the upper-critical dimension $d=6$. Indeed, we show in particular that if the triangle diagram diverges polylogarithmically as $p \uparrow p_c$ then mean-field critical behaviour holds to within a polylogarithmic factor. We apply the methods we develop to deduce that for long-range percolation on the hierarchical lattice, mean-field critical behaviour holds to within polylogarithmic factors at the upper-critical dimension. As part of the proof, we introduce a new method for comparing diagrammatic sums on general transitive graphs that may be of independent interest.