The Derivation of the Lace Expansion via Inclusion–Exclusion

Abstract

We perform the combinatorial expansion that relies on inclusion-exclusion for the percolation two-point function \(\tau _p(x)=\mathbb {P}_{p}(0 \leftrightarrow x)\) that goes under the name lace expansion. We give a self-contained and detailed derivation of this expansion, and close this chapter by discussing how it can be effectively used, focussing in particular on the error term that arises in the inclusion-exclusion expansion.