TY - JOUR

T1 - Expansion in $n^{-1}$ for percolation critical values on the n-cube and $Z^n$ : the first three terms

AU - Hofstad, van der, R.W.

AU - Slade, G.

PY - 2006

Y1 - 2006

N2 - 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 ${\mathbb Z}^n$.

AB - 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 ${\mathbb Z}^n$.

U2 - 10.1017/S0963548306007498

DO - 10.1017/S0963548306007498

M3 - Article

VL - 15

SP - 695

EP - 713

JO - Combinatorics, Probability and Computing

JF - Combinatorics, Probability and Computing

SN - 0963-5483

IS - 5

ER -