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- W3135969498 abstract "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, where p+q≤1. 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(logp−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." @default.
- W3135969498 created "2021-03-29" @default.
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- W3135969498 date "2021-02-01" @default.
- W3135969498 modified "2023-10-02" @default.
- W3135969498 title "Polluted bootstrap percolation in three dimensions" @default.
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- W3135969498 doi "https://doi.org/10.1214/20-aap1588" @default.
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