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 ∇_(p_c) is large. In contrast to earlier methods, our approach continues to yield bounds of reasonable order when the triangle diagram ∇^p is unbounded but diverges slowly as p ↑ p_c, as is expected to occur in percolation on ℤ^d at the upper-critical dimension d=6. Indeed, we show in particular that if the triangle diagram diverges polylogarithmically as p↑pc 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.