Scaling limit and critical exponents for two-dimensional bootstrap percolation

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Abstract

Consider a cellular automaton with state space {0, 1}Z2 where the initial configuration ω0 is chosen according to a Bernoulli product measure, 1’s are stable, and 0’s become 1’s if they are surrounded by at least three neighboring 1’s. In this paper we show that the configuration ωn at time n converges exponentially fast to a final configuration. (Formula Presented.), and that the limiting measure corresponding to (Formula Presented.) is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents β, η, ν and γ, and of the continuum scaling limit of crossing probabilities for independent site percolation on the close-packed version of Z2 (i.e. for independent ∗-percolation on Z2), 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.

Original languageEnglish (US)
Pages (from-to)85-101
Number of pages17
JournalJournal of Statistical Physics
Volume118
Issue number1-2
DOIs
StatePublished - Jan 1 2005

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Keywords

  • Bootstrap percolation
  • Critical exponents
  • Scaling limit
  • Universality

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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