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• W3169963873 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." @default.
• W3169963873 created "2021-06-22" @default.
• W3169963873 creator A5060156386 @default.
• W3169963873 date "2021-06-11" @default.
• W3169963873 modified "2023-09-27" @default.
• W3169963873 title "On the derivation of mean-field percolation critical exponents from the triangle condition" @default.
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• W3169963873 hasPublicationYear "2021" @default.
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