Polluted bootstrap percolation with threshold two in all dimensions

Janko Gravner, Alexander Holroyd

Research output: Contribution to journalArticle (Academic Journal)peer-review


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 Z^d of dimension d≥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.
Original languageEnglish
Pages (from-to)467–486
Number of pages19
JournalProbability Theory and Related Fields
Issue number1-2
Early online date18 Dec 2018
Publication statusPublished - 1 Oct 2019

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