Three-dimensional 2-critical bootstrap percolation: The stable sets approach

TL;DR

This paper analyzes the critical conditions for percolation in a 3D bootstrap model with varying infection thresholds, providing precise asymptotic estimates for the critical length in different parameter regimes.

Contribution

It determines the logarithm of the critical length for percolation in a 3D bootstrap percolation model across a broad parameter range, extending previous results.

Findings

Exact critical length estimates for certain parameter ranges

Upper bounds for remaining cases

Insights into the percolation threshold behavior

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 $v\in [L]^3$ already having $r$ infected neighbours, and that infected sites remain infected forever. In this paper we determine $\log$ of the critical length for percolation up to a constant factor, for all $r\in \{a_3+1, \dots, a_3+a_2\}$ with $a_3\ge a_1+a_2$. We moreover give upper bounds for all remaining cases $a_3 < a_1+a_2$ and believe that they are tight up to a constant factor.