Expansion in ${\boldsymbol{n^{-1}}}$ for Percolation Critical Values on the $n$-cube and ${\boldsymbol{{\mathbb Z}^n}}$: 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 ${\mathbb G} = {\mathbb Z}^n$, \[p_c({\mathbb G}) = \Omega^{-1} + \Omega^{-2} + \frac{7}{2} \Omega^{-3} + O(\Omega^{-4}).\] This extends by two terms the result $p_c({\mathbb Q}_n) = \Omega^{-1} + O(\Omega^{-2})$ of Borgs, Chayes, van der Hofstad, Slade and Spencer, and provides a simplified proof of a previous result of Hara and Slade for ${\mathbb Z}^n$.