A generalised inductive approach to the lace expansion

TL;DR

This paper develops a generalized inductive method to analyze the lace expansion, providing conditions for Gaussian behavior in models like self-avoiding walks and percolation, aiding understanding of their critical phenomena.

Contribution

It introduces a unified inductive framework for the lace expansion, applicable to various high-dimensional models, and establishes conditions for Gaussian asymptotics.

Findings

Conditions for Gaussian behavior established

Applicable to high-dimensional self-avoiding walks and percolation

Provides a foundation for detailed critical behavior analysis

Abstract

The lace expansion is a powerful tool for analysing the critical behaviour of self-avoiding walks and percolation. It gives rise to a recursion relation which we abstract and study using an adaptation of the inductive method introduced by den Hollander and the authors. We give conditions under which the solution to the recursion relation behaves as a Gaussian, both in Fourier space and in terms of a local central limit theorem. These conditions are shown elsewhere to hold for sufficiently spread-out models of networks of self-avoiding walks in dimensions $d>4$, and for critical oriented percolation in dimensions $d+1>5$, providing a unified approach and an essential ingredient for a detailed analysis of the branching behaviour of these models.