Scaling limit and critical exponents for two-dimensional bootstrap percolation

F. Camia

    Research output: Contribution to journalArticleAcademicpeer-review

    2 Citations (Scopus)

    Abstract

    Consider a cellular automaton with state space {0,1} 2 where the initial configuration _0 is chosen according to a Bernoulli product measure, 1s are stable, and 0s become 1s if they are surrounded by at least three neighboring 1s. In this paper we show that the configuration _n at time n converges exponentially fast to a final configuration , and that the limiting measure corresponding to is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents , , and , and of the continuum scaling limit of crossing probabilities for independent site percolation on the close-packed version of 2 (i.e. for independent *-percolation on ), we prove that the bootstrapped percolation model has the same scaling limit and critical exponents.This type of bootstrap percolation can be seen as a paradigm for a class of cellular automata whose evolution is given, at each time step, by a monotonic and nonessential enhancement [Aizenman and Grimmett, J. Stat. Phys. 63: 817--835 (1991); Grimmett, Percolation, 2nd Ed. (Springer, Berlin, 1999)
    Original languageEnglish
    Pages (from-to)85-101
    JournalJournal of Statistical Physics
    Volume118
    Issue number1-2
    DOIs
    Publication statusPublished - 2005

    Fingerprint Dive into the research topics of 'Scaling limit and critical exponents for two-dimensional bootstrap percolation'. Together they form a unique fingerprint.

  • Cite this