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.
Original language  English 

Pages (fromto)  467–486 
Number of pages  19 
Journal  Probability Theory and Related Fields 
Volume  175 
Issue number  12 
Early online date  18 Dec 2018 
DOIs  
Publication status  Published  1 Oct 2019 
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Profiles

Professor Alexander E Holroyd
Person: Academic