Random majority percolation

  • Authors:

  • Paul Balister;Béla Bollobás;J. Robert Johnson;Mark Walters

  • Affiliations:

  • Department of Mathematics, University of Memphis, Memphis, Tennessee 38152;Department of Mathematics, University of Memphis, Memphis, Tennessee 38152 and Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, UK;Queen Mary University of London, London E1 4NS, UK;Queen Mary University of London, London E1 4NS, UK

  • Venue:
  • Random Structures & Algorithms
  • Year:
  • 2010

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Abstract

We shall consider the discrete time synchronous random majority-vote cellular automata on the n by n torus, in which every vertex is in one of two states and, at each time step t, every vertex goes into the state the majority of its neighbors had at time t - 1 with a small chance p of error independently of all other events. We shall show that, if n is fixed and p is sufficiently small, then the process spends almost half of its time in each of two configurations. Further more, we show that the expected time for it to reach one of these configurations from the other is Θ(1-pn+1) despite the actual time spent in transit being O(1-p3). Unusually, it appears difficult to obtain any results for this regime by simulation. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010