Open Access
September 2020 Anisotropic bootstrap percolation in three dimensions
Daniel Blanquicett
Ann. Probab. 48(5): 2591-2614 (September 2020). DOI: 10.1214/20-AOP1434


Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^{3}$, and assume that the neighborhood of each vertex consists of the $a_{i}$ nearest neighbors in the $\pm e_{i}$-directions for each $i\in \{1,2,3\}$, where $a_{1}\le a_{2}\le a_{3}$. Suppose we infect any healthy vertex $x\in [L]^{3}$ already having $a_{3}+1$ infected neighbors, and that infected sites remain infected forever. In this paper, we determine the critical length for percolation up to a constant factor in the exponent, for all triples $(a_{1},a_{2},a_{3})$. To do so, we introduce a new algorithm called the beams process and prove an exponential decay property for a family of subcritical two-dimensional bootstrap processes.


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Daniel Blanquicett. "Anisotropic bootstrap percolation in three dimensions." Ann. Probab. 48 (5) 2591 - 2614, September 2020.


Received: 1 September 2019; Published: September 2020
First available in Project Euclid: 23 September 2020

MathSciNet: MR4152652
Digital Object Identifier: 10.1214/20-AOP1434

Primary: 60K35
Secondary: 60C05

Keywords: Anisotropic bootstrap percolation , beams process , Exponential decay

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 5 • September 2020
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