Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation
| dc.creator | Holroyd, Alexander E. | |
| dc.date | 2002-06-12 | |
| dc.date.accessioned | 2026-07-07T04:49:06Z | |
| dc.date.available | 2026-07-07T04:49:06Z | |
| dc.description | In the bootstrap percolation model, sites in an $L$ by $L$ square are initially independently declared active with probability $p$. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour as $p \to 0$ and $L \to \infty$ simultaneously of the probability $I(L,p)$ that the entire square is eventually active. We prove that $I(L,p) \to 1$ if $\liminf p \log L > λ$, and $I(L,p) \to 0$ if $\limsup p \log L < λ$, where $λ= π^2/18$. We prove the same behaviour, with the same threshold $λ$, for the probability $J(L,p)$ that a site is active by time $L$ in the process on the infinite lattice. The same results hold for the so-called modified bootstrap percolation model, but with threshold $λ' = π^2/6$. The existence of the thresholds $λ,λ'$ settles a conjecture of Aizenman and Lebowitz, while the determination of their values corrects numerical predictions of Adler, Stauffer and Aharony. | |
| dc.identifier | https://arxiv.org/abs/math/0206132 | |
| dc.identifier | http://arxiv.org/abs/math/0206132 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/64294 | |
| dc.subject | Probability | |
| dc.subject | Mathematical Physics | |
| dc.subject | 60K35; 82B43 | |
| dc.title | Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation | |
| dc.type | text |