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

TL;DR

This paper introduces a novel method to derive mean-field exponents for spread-out Bernoulli percolation in high dimensions, providing bounds for two-point functions in critical regimes.

Contribution

It presents a new approach for analyzing mean-field behavior in high-dimensional percolation, extending understanding beyond traditional methods.

Findings

Derived upper bounds for two-point functions in full and half-spaces

Applicable to critical and near-critical regimes

Supports analysis of related models like self-avoiding walk

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 $d>4$.