Scaling and Inverse Scaling in Anisotropic Bootstrap percolation

TL;DR

This paper investigates the behavior of critical thresholds in anisotropic bootstrap percolation, focusing on higher-order corrections and inverse scaling, which are challenging to determine in general but can be simplified in specific 2D cases.

Contribution

It provides new insights into higher-order correction terms for percolation thresholds in anisotropic bootstrap models, especially in two dimensions.

Findings

Higher-order correction terms can be derived from inversion in 2D anisotropic models.

Critical thresholds converge slowly to zero with system size.

Inverse scaling behavior is characterized for specific anisotropic cases.

Abstract

In bootstrap percolation it is known that the critical percolation threshold tends to converge slowly to zero with increasing system size, or, inversely, the critical size diverges fast when the percolation probability goes to zero. To obtain higher-order terms (that is, sharp and sharper thresholds) for the percolation threshold in general is a hard question. In the case of two-dimensional anisotropic models, sometimes correction terms can be obtained from inversion in a relatively simple manner.