Slow Convergence in Bootstrap Percolation
| dc.creator | Gravner, Janko | |
| dc.creator | Holroyd, Alexander E. | |
| dc.date | 2007-05-09 | |
| dc.date.accessioned | 2026-07-07T08:00:26Z | |
| dc.date.available | 2026-07-07T08:00:26Z | |
| dc.description | In the bootstrap percolation model, sites in an L by L square are initially infected independently with probability p. At subsequent steps, a healthy site becomes infected if it has at least 2 infected neighbours. As (L,p)->(infinity,0), the probability that the entire square is eventually infected is known to undergo a phase transition in the parameter p log L, occurring asymptotically at lambda = pi^2/18. We prove that the discrepancy between the critical parameter and its limit lambda is at least Omega((log L)^(-1/2)). In contrast, the critical window has width only Theta((log L)^(-1)). For the so-called modified model, we prove rigorous explicit bounds which imply for example that the relative discrepancy is at least 1% even when L = 10^3000. Our results shed some light on the observed differences between simulations and rigorous asymptotics. | |
| dc.description | 22 pages, 3 figures | |
| dc.identifier | https://arxiv.org/abs/0705.1347 | |
| dc.identifier | http://arxiv.org/abs/0705.1347 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/128667 | |
| dc.subject | Probability | |
| dc.subject | Mathematical Physics | |
| dc.subject | 60K35; 82B43 | |
| dc.title | Slow Convergence in Bootstrap Percolation | |
| dc.type | text |