Metastability threshold for anisotropic bootstrap percolation in three dimensions

TL;DR

This paper investigates the metastability thresholds in three-dimensional anisotropic bootstrap percolation models, revealing that thresholds are influenced by the easiest growth direction and can match isotropic models.

Contribution

It provides the order of magnitude for metastability thresholds in a broad class of 3D anisotropic bootstrap percolation models using dimensional reduction techniques.

Findings

Thresholds are determined by the easiest growth direction.

In 3D, thresholds can equal those of isotropic models.

The study extends understanding of anisotropic bootstrap percolation behavior.

Abstract

In this paper we analyze several anisotropic bootstrap percolation models in three dimensions. We present the order of magnitude for the metastability threshold for a fairly general class of models. In our proofs we use an adaptation of the technique of dimensional reduction. We find that the order of the metastability threshold is generally determined by the "easiest growth direction" in the model. In contrast to the anisotropic bootstrap percolation in two dimensions, in three dimensions the order of the metatstability threshold for anisotropic bootstrap percolation can be equal to that of isotropic bootstrap percolation.