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

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.

36 pages, 3 figures. V2: Fixed a minor error. V3: Various minor fixes and improvements; content reorganized. Accepted version, to appear in JSP

References in corpus (3)

• Random subgraphs of finite graphs: I. The scaling window under the triangle condition
• Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk
• The critical two-point function for long-range percolation on the hierarchical lattice

Cited by in corpus (1)

• Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on ${\mathbb{Z}}^d$