Polluted bootstrap percolation with threshold two in all dimensions

  • Janko Gravner
  • Alexander E. HolroydEmail author


In the polluted bootstrap percolation model, the vertices of a graph are independently declared initially occupied with probability p or closed with probability q. At subsequent steps, a vertex becomes occupied if it is not closed and it has at least r occupied neighbors. On the cubic lattice \({\mathbb {Z}}^d\) of dimension \(d\ge 3\) with threshold \(r=2\), we prove that the final density of occupied sites converges to 1 as p and q both approach 0, regardless of their relative scaling. Our result partially resolves a conjecture of Morris, and contrasts with the \(d=2\) case, where Gravner and McDonald proved that the critical parameter is \(q/{p^2}\).


Bootstrap percolation Cellular automaton Critical scaling 

Mathematics Subject Classification

60K35 82B43 



We thank David Sivakoff for many valuable discussions. Janko Gravner was partially supported by the NSF grant DMS–1513340, Simons Foundation Award #281309, and the Slovenian Research Agency program P1–285. He also gratefully acknowledges the hospitality of the Theory Group at Microsoft Research, where most of this work was completed. We thank the anonymous referee for helpful comments.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of CaliforniaDavisUSA
  2. 2.University of WashingtonSeattleUSA

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