Two Phase Transitions in Two-Way Bootstrap Percolation

Author Ahad N. Zehmakan

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Ahad N. Zehmakan
  • ETH Zurich, Switzerland


The author likes to thank Raphael Cerf, Bernd Gärtner, and Roberto H. Schonmann for several stimulating discussions.

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Ahad N. Zehmakan. Two Phase Transitions in Two-Way Bootstrap Percolation. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 5:1-5:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Consider a graph G and an initial random configuration, where each node is black with probability p and white otherwise, independently. In discrete-time rounds, each node becomes black if it has at least r black neighbors and white otherwise. We prove that this basic process exhibits a threshold behavior with two phase transitions when the underlying graph is a d-dimensional torus and identify the threshold values.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • bootstrap percolation
  • cellular automata
  • phase transition
  • d-dimensional torus
  • r-threshold model
  • biased majority


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