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

András Bálint, Federico Camia, Ronald Meester

Research output: Contribution to journalArticlepeer-review

Abstract

We study a natural dependent percolation model introduced by Häggström. Consider subcritical Bernoulli bond percolation with a fixed parameter p < pc. 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, rc (p) and rc* (p) respectively, as usual, we prove that rc (p) + rc* (p) = 1 for all subcritical p. On the triangular lattice, where our method also works, this leads to rc (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 rc (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.

Original languageEnglish (US)
Pages (from-to)937-965
Number of pages29
JournalStochastic Processes and their Applications
Volume119
Issue number3
DOIs
StatePublished - Mar 2009

Keywords

  • Critical behaviour
  • DaC model
  • Dependent percolation
  • Duality
  • RSW theorem
  • Sharp phase transition
  • p = 1 / 2

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics

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