Higher order corrections for anisotropic bootstrap percolation

TL;DR

This paper refines the asymptotic estimates of the critical probability for a specific anisotropic bootstrap percolation model, revealing that higher order terms significantly impact the approximation even for extremely large lattices.

Contribution

The authors derive sharp second and third order asymptotics for the critical probability in anisotropic bootstrap percolation, improving upon previous first order results.

Findings

Second and third order asymptotics explicitly derived

Higher order terms dominate first order approximation for large lattices

First order asymptotics are inadequate even for unimaginably large sizes

Abstract

We study the critical probability for the metastable phase transition of the two-dimensional anisotropic bootstrap percolation model with $(1,2)$-neighbourhood and threshold $r = 3$. The first order asymptotics for the critical probability were recently determined by the first and second authors. Here we determine the following sharp second and third order asymptotics: \[ p_c\big( [L]^2,\mathcal{N}_{(1,2)},3 \big) \; = \; \frac{(\log \log L)^2}{12\log L} \, - \, \frac{\log \log L \, \log \log \log L}{ 3\log L} + \frac{\left(\log \frac{9}{2} + 1 \pm o(1) \right)\log \log L}{6\log L}. \] We note that the second and third order terms are so large that the first order asymptotics fail to approximate $p_c$ even for lattices of size well beyond $10^{10^{1000}}$.