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

Daniel Blanquicett

paper

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

arXiv:2201.11365

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 $a_{1} \leq a_{2} \leq a_{3}$ . 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 $lo g$ of the critical length for percolation up to a constant factor, for all $r \in {a_{3} + 1, \dots, a_{3} + a_{2}}$ with $a_{3} \geq a_{1} + a_{2}$ . 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.

arXiv admin note: text overlap with arXiv:2201.09029