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

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 $$\mathbb{Q}$_n = {0,1}^n$andon$\Z^n$,respectively\mathrm{.}Let$\Omega = n$for$$\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 ${\mathbb{Z}}^n$.