An alternative approach for the mean-field behaviour of spread-out Bernoulli percolation in dimensions $d>6$

Abstract

This article proposes a new way of deriving mean-field exponents for sufficiently spread-out Bernoulli percolation in dimensions $d>6$. We obtain an upper bound for the full-space and half-space two-point functions in the critical and near-critical regimes. In a companion paper, we apply a similar analysis to the study of the weakly self-avoiding walk model in dimensions $d>4$.

60 pages, 7 figures. We added the derivation of lower bounds in Theorem 1.3. Accepted version, to appear in Probability Theory and Related Fields