Critical percolation on random regular graphs
| dc.creator | Nachmias, Asaf | |
| dc.creator | Peres, Yuval | |
| dc.date | 2007-07-19 | |
| dc.date | 2007-07-20 | |
| dc.date.accessioned | 2026-07-07T08:19:20Z | |
| dc.date.available | 2026-07-07T08:19:20Z | |
| dc.description | We describe the component sizes in critical independent p-bond percolation on a random d-regular graph on n vertices, where d \geq 3 is fixed and n grows. We prove mean-field behavior around the critical probability p_c=1/(d-1). In particular, we show that there is a scaling window of width n^{-1/3} around p_c in which the sizes of the largest components are roughly n^{2/3} and we describe their limiting joint distribution. We also show that for the subcritical regime, i.e. p = (1-eps(n))p_c where eps(n)=o(1) but \eps(n)n^{1/3} tends to infinity, the sizes of the largest components are concentrated around an explicit function of n and eps(n) which is of order o(n^{2/3}). In the supercritical regime, i.e. p = (1+\eps(n))p_c where eps(n)=o(1) but eps(n)n^{1/3} tends to infinity, the size of the largest component is concentrated around the value (2d/(d-2))\eps(n)n and a duality principle holds: other component sizes are distributed as in the subcritical regime. | |
| dc.description | 47 pages | |
| dc.identifier | https://arxiv.org/abs/0707.2839 | |
| dc.identifier | http://arxiv.org/abs/0707.2839 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/134727 | |
| dc.subject | Probability | |
| dc.subject | Combinatorics | |
| dc.title | Critical percolation on random regular graphs | |
| dc.type | text |