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A Sharp Threshold for a Modified Bootstrap Percolation with Recovery

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Abstract

Bootstrap percolation is a type of cellular automaton on graphs, introduced as a simple model of the dynamics of ferromagnetism. Vertices in a graph can be in one of two states: ‘healthy’ or ‘infected’ and from an initial configuration of states, healthy vertices become infected by local rules. While the usual bootstrap processes are monotone in the sets of infected vertices, in this paper, a modification is examined in which infected vertices can return to a healthy state. Vertices are initially infected independently at random and the central question is whether all vertices eventually become infected. The model examined here is such a process on a square grid for which healthy vertices with at least two infected neighbours become infected and infected vertices with no infected neighbours become healthy. Sharp thresholds are given for the critical probability of initial infections for all vertices eventually to become infected.

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Author notes

  1. Karen Gunderson

    Present address: Heilbronn Institute for Mathematical Research, School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

Authors and Affiliations

  1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK

    Tom Coker

  2. Department of Mathematical Sciences, University of Memphis, Memphis, TN, 38152, USA

    Karen Gunderson

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  1. Tom Coker
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  2. Karen Gunderson
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Correspondence to Karen Gunderson.

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Coker, T., Gunderson, K. A Sharp Threshold for a Modified Bootstrap Percolation with Recovery. J Stat Phys 157, 531–570 (2014). https://doi.org/10.1007/s10955-014-1099-3

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  • DOI: https://doi.org/10.1007/s10955-014-1099-3

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