Sharp phase transition and critical behaviour in 2D divide and colour models
| dc.creator | Balint, Andras | |
| dc.creator | Camia, Federico | |
| dc.creator | Meester, Ronald | |
| dc.date | 2007-08-24 | |
| dc.date.accessioned | 2026-07-07T08:25:35Z | |
| dc.date.available | 2026-07-07T08:25:35Z | |
| dc.description | Consider subcritical Bernoulli bond percolation with fixed parameter p<p_c. We define a dependent site percolation model by the following procedure: for each bond cluster, we colour all vertices in the cluster black with probability r and white with probability 1-r, independently of each other. On the square lattice, defining the critical probabilities for the site model and its dual, r_c(p) and r_c^*(p) respectively, as usual, we prove that r_c(p)+r_c^*(p)=1 for all subcritical p. On the triangular lattice, where our method also works, this leads to r_c(p)=1/2, for all subcritical p. On both lattices, we obtain exponential decay of cluster sizes below r_c(p), divergence of the mean cluster size at r_c(p), and continuity of the percolation function in r on [0,1]. We also discuss possible extensions of our results, and formulate some natural conjectures. Our methods rely on duality considerations and on recent extensions of the classical RSW theorem. | |
| dc.description | 39 pages, 4 figures | |
| dc.identifier | https://arxiv.org/abs/0708.3349 | |
| dc.identifier | http://arxiv.org/abs/0708.3349 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/136669 | |
| dc.subject | Probability | |
| dc.subject | Mathematical Physics | |
| dc.subject | 60K35, 82B43, 82B20 | |
| dc.title | Sharp phase transition and critical behaviour in 2D divide and colour models | |
| dc.type | text |