The second largest component in the supercritical 2D Hamming graph
| dc.creator | van der Hofstad, Remco | |
| dc.creator | Luczak, Malwina J. | |
| dc.creator | Spencer, Joel | |
| dc.date | 2008-01-10 | |
| dc.date | 2009-01-05 | |
| dc.date.accessioned | 2026-07-07T12:23:32Z | |
| dc.date.available | 2026-07-07T12:23:32Z | |
| dc.description | The 2-dimensional Hamming graph H(2,n) consists of the $n^2$ vertices $(i,j)$, $1\leq i,j\leq n$, two vertices being adjacent when they share a common coordinate. We examine random subgraphs of H(2,n) in percolation with edge probability $p$, so that the average degree $2(n-1)p=1+ε$. Previous work by van der Hofstad and Luczak had shown that in the barely supercritical region $n^{-2/3}\ln^{1/3}n\ll ε\ll 1$ the largest component has size $\sim 2εn$. Here we show that the second largest component has size close to $ε^{-2}$, so that the dominant component has emerged. This result also suggests that a {\it discrete duality principle} might hold, whereby, after removing the largest connected component in the supercritical regime, the remaining random subgraphs behave as in the subcritical regime. | |
| dc.description | 9 pages, revised version | |
| dc.identifier | https://arxiv.org/abs/0801.1608 | |
| dc.identifier | http://arxiv.org/abs/0801.1608 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/214023 | |
| dc.subject | Probability | |
| dc.subject | Combinatorics | |
| dc.subject | 05C80 | |
| dc.title | The second largest component in the supercritical 2D Hamming graph | |
| dc.type | text |