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

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.