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arXiv · 1705.01652

Polluted Bootstrap Percolation with Threshold Two in All Dimensions

Janko Gravner·Alexander E. Holroyd
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Abstract

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}.

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BibTeXRIS

Janko Gravner, Alexander E. Holroyd. 2017-05-03. Polluted Bootstrap Percolation with Threshold Two in All Dimensions. https://arxiv.org/abs/1705.01652

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