Expansion in $n^{-1}$ for percolation critical values on the $n$-cube and $Z^n$: the first three terms
| dc.creator | van der Hofstad, Remco | |
| dc.creator | Slade, Gordon | |
| dc.date | 2004-01-08 | |
| dc.date.accessioned | 2026-07-07T06:32:56Z | |
| dc.date.available | 2026-07-07T06:32:56Z | |
| dc.description | 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 $\Z^n$, respectively. Let $Ω= n$ for $\mathbb{G} = \mathbb{Q}_n$ and $Ω= 2n$ 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 $\mathbb{Z}^n$. | |
| dc.description | 18 pages, 3 figures | |
| dc.identifier | https://arxiv.org/abs/math/0401072 | |
| dc.identifier | http://arxiv.org/abs/math/0401072 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/99027 | |
| dc.subject | Probability | |
| dc.subject | Combinatorics | |
| dc.subject | 05C80, 60K35, 82B43 | |
| dc.title | Expansion in $n^{-1}$ for percolation critical values on the $n$-cube and $Z^n$: the first three terms | |
| dc.type | text |