An extension of the inductive approach to the lace expansion

TL;DR

This paper broadens the inductive approach to the lace expansion, enabling its application to a wider range of models, and demonstrates its utility in analyzing high-dimensional lattice trees and percolation.

Contribution

It extends the inductive method for the lace expansion beyond critical dimension 4, facilitating analysis of models in higher dimensions.

Findings

Proved Gaussian asymptotics for lattice trees in dimensions d>8.

Potential applicability to percolation in dimensions d>6.

Enhanced analytical tools for high-dimensional probabilistic models.

Abstract

We extend the inductive approach to the lace expansion, previously developed to study models with critical dimension 4, to be applicable more generally. In particular, the result of this note has recently been used to prove Gaussian asymptotic behaviour for the Fourier transform of the two-point function for sufficiently spread-out lattice trees in dimensions d>8, and it is potentially also applicable to percolation in dimensions d>6.