Anisotropic bootstrap percolation in three dimensions

TL;DR

This paper investigates the critical size for percolation in a three-dimensional anisotropic bootstrap process, introducing a new algorithm and proving exponential decay properties for subcritical cases.

Contribution

It determines the critical length for percolation in 3D anisotropic bootstrap percolation for all parameter triples, using a novel beams process algorithm.

Findings

Critical length for percolation up to a constant factor in the exponent.

Introduction of the beams process algorithm.

Exponential decay property for subcritical bootstrap processes.

Abstract

Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^3$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours 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 neighbours, 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.