The $d$-dimensional bootstrap percolation models with threshold at least double exponential

TL;DR

This paper precisely determines the critical length scale for percolation in high-dimensional bootstrap models with complex neighborhood structures, revealing the iterated logarithm behavior up to a constant factor.

Contribution

It establishes the asymptotic critical length for bootstrap percolation in arbitrary dimensions with general neighborhood parameters, extending previous results to more complex models.

Findings

Critical length characterized by iterated logarithm of system size.

Reduces threshold determination to lower-dimensional cases for all dimensions.

Provides bounds valid up to a constant factor for all parameters.

Abstract

Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^d$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $\pm e_i$-directions for each $i \in \{1,2,\dots, d\}$, where $a_1\le a_2\le \dots \le a_d$. Suppose we infect any healthy vertex $v\in [L]^d$ already having $r$ infected neighbours, and that infected sites remain infected forever. In this paper we determine the $(d-1)$-times iterated logarithm of the critical length for percolation up to a constant factor, for all $d$-tuples $(a_1,\dots ,a_d)$ and all $r\in \{a_2+\dots + a_d+1, \dots, a_1+a_2+\dots + a_d\}$. Moreover, we reduce the problem of determining this (coarse) threshold for all $d\ge 3$ and all $r\in \{a_d+1, \dots, a_1+a_2+\dots + a_d\}$, to that of determining the threshold for all $d\ge 3$ and all $r\in \{ a_d+1, \dots, a_{d-1} + a_d\}$.