2017/06/22 by Gravner, Janko, Holroyd, Alexander E., Sivakoff, David · 2 citations
#60K35 #82B43 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1706.07338
In the polluted bootstrap percolation model, vertices of the cubic lattice ℤ3 are independently declared initially occupied with probability p or closed with probability q. Under the standard (respectively, modified) bootstrap rule, a vertex becomes occupied at a subsequent step if it is not closed and it has at least 3 occupied neighbors (respectively, an occupied neighbor in each coordinate). We study the final density of occupied vertices as p,q→ 0. We show that this density converges to 1 if q ≪ p3(log p-1)-3 for both standard and modified rules. Our principal result is a complementary bound with a matching power for the modified model: there exists C such that the final density converges to 0 if q > Cp3. For the standard model, we establish convergence to 0 under the stronger condition q>Cp2.