Expansion in $n^{-1}$ for percolation critical values on the $n$-cube and $Z^n$: the first three terms

Abstract

Let $p_c({\mathbb{Q}}_n)$ and $p_c({\mathbb{Z}}^n)$ denote the critical values for nearest-neighbour bond percolation on the $n$-cube ${\mathbb{Q}}_n=\{0,1{\}}^n$ and on ${\mathbb{Z}}^n$, respectively. Let $\Omega =n$ for $\mathbb{G}={\mathbb{Q}}_n$ and $\Omega =2n$ for $\mathbb{G}={\mathbb{Z}}^n$ denote the degree of $\mathbb{G}$. We use the lace expansion to prove that for both $\mathbb{G}={\mathbb{Q}}_n$ and $\mathbb{G}={\mathbb{Z}}^n$, $p_c(\mathbb{G}) & = \cn^{-1} + \cn^{-2} + {7/2} \cn^{-3} + O(\cn^{-4}).$ This extends by two terms the result $p_c(\mathbb{Q}_n) = \cn^{-1} + O(\cn^{-2})$ of Borgs, Chayes, van der Hofstad, Slade and Spencer, and provides a simplified proof of a previous result of Hara and Slade for ${\mathbb{Z}}^n$.

18 pages, 3 figures

References in corpus (1)

• Random subgraphs of finite graphs: I. The scaling window under the triangle condition