A Survey on the Lace Expansion for the Nearest-neighbor Models on the BCC Lattice

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.