A survey on the lace expansion for the nearest-neighbor models on the BCC lattice

TL;DR

This survey explains the lace expansion technique for self-avoiding walk and percolation models on the BCC lattice, making the complex analysis accessible for dimensions above 6 and 9.

Contribution

It provides a beginner-friendly, self-contained explanation of the lace expansion on the BCC lattice, highlighting its convergence in higher dimensions.

Findings

Lace expansion converges for models above certain dimensions

Enumeration of random-walk quantities is simplified on BCC lattice

A specific set of bootstrapping functions clarifies the analysis

Abstract

The aim of this survey is to explain, in a self-contained and relatively beginner-friendly manner, the lace expansion for the nearest-neighbor models of self-avoiding walk and percolation that converges in all dimensions above 6 and 9, respectively. To achieve this, we consider a $d$-dimensional version of the body-centered cubic (BCC) lattice, on which it is extremely easy to enumerate various random-walk quantities. Also, we choose a particular set of bootstrapping functions, by which a notoriously complicated part of the lace-expansion analysis becomes rather transparent.