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 diagram $\nabla_{p_c}$ large. In contrast to earlier methods, our approach continues yield bounds reasonable order $\nabla_p$ unbounded but diverges slowly as $p \uparrow p_c$, expected occur in on $\mathbb{Z}^d$ at upper-critical dimension $d=6$. Indeed, we show particular if polylogarithmically p_c$ then holds within polylogarithmic factor. apply methods develop deduce for long-range hierarchical lattice, factors dimension. As part proof, introduce method comparing diagrammatic sums general transitive graphs may be independent interest.