Sharp phase transition and critical behaviour in 2D divide and colour models

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:119

Article

Meester, Ronald¹ 

[1] Vrije Univ Amsterdam, Sect Stochastiek, Div Wiskunde, Fac Exacte Wetenschappen,Dept Math, NL-1081 HV Amsterdam, Netherlands

关键词: Dependent percolation;    Sharp phase transitions;    Critical behaviour;    Duality;    DaC model;    RSW theorem;    p(c)=1/2;   

DOI  :  10.1016/j.spa.2008.04.003

来源: Elsevier

【 摘 要 】

We study a natural dependent percolation model introduced by Haggstrom. Consider subcritical Bernoulli bond percolation with a 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 tor 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 rc(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 lorInUlate sorne, natural conjectures. Our methods rely on duality considerations and On recent extensions ot'the classical RSW theorem. (C) 2008 Elsevier B.V. All rights reserved.

【 授权许可】

Free