Open Access
February 2021 Polluted bootstrap percolation in three dimensions
Janko Gravner, Alexander E. Holroyd, David Sivakoff
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Ann. Appl. Probab. 31(1): 218-246 (February 2021). DOI: 10.1214/20-AAP1588


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+q1. 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,q0. We show that this density converges to 1 if qp3(logp1)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.


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Janko Gravner. Alexander E. Holroyd. David Sivakoff. "Polluted bootstrap percolation in three dimensions." Ann. Appl. Probab. 31 (1) 218 - 246, February 2021.


Received: 1 June 2019; Published: February 2021
First available in Project Euclid: 8 March 2021

Digital Object Identifier: 10.1214/20-AAP1588

Primary: 60K35 , 82B43

Keywords: Bootstrap percolation , cellular automaton , critical scaling

Rights: Copyright © 2021 Institute of Mathematical Statistics

Vol.31 • No. 1 • February 2021
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