An alternative approach for the mean-field behaviour of weakly self-avoiding walks in dimensions $d>4$

TL;DR

This paper introduces a novel method to derive mean-field exponents for weakly self-avoiding walks in high dimensions, providing precise estimates for two-point functions in critical regimes.

Contribution

It presents a new approach for analyzing mean-field behavior in self-avoiding walks, extending understanding in dimensions greater than four.

Findings

Derived up-to-constant estimates for two-point functions

Established results in critical and near-critical regimes

Proposed a methodology applicable to high-dimensional models

Abstract

This article proposes a new way of deriving mean-field exponents for the weakly self-avoiding walk model in dimensions $d>4$. Among other results, we obtain up-to-constant estimates for the full-space and half-space two-point functions in the critical and near-critical regimes. A companion paper proposes a similar analysis for spread-out Bernoulli percolation in dimensions $d>6$.