Universality class for bootstrap percolation with $m=3$ on the cubic lattice

TL;DR

This study uses Monte Carlo simulations to analyze the $m=3$ bootstrap percolation on a cubic lattice, revealing it shares the same universality class as standard percolation, contrary to previous beliefs.

Contribution

The paper demonstrates that the $m=3$ bootstrap percolation model on a cubic lattice belongs to the same universality class as ordinary percolation, using finite-size scaling and numerical analysis.

Findings

The critical probability $p_c$ aligns with usual percolation.

The critical spanning probability $R(p_c)$ matches that of standard percolation.

The model exhibits the same scaling powers as standard percolation.

Abstract

We study the $m=3$ bootstrap percolation model on a cubic lattice, using Monte Carlo simulation and finite-size scaling techniques. In bootstrap percolation, sites on a lattice are considered occupied (present) or vacant (absent) with probability $p$ or $1-p$, respectively. Occupied sites with less than $m$ occupied first-neighbours are then rendered unoccupied; this culling process is repeated until a stable configuration is reached. We evaluate the percolation critical probability, $p_c$, and both scaling powers, $y_p$ and $y_h$, and, contrarily to previous calculations, our results indicate that the model belongs to the same universality class as usual percolation (i.e., $m=0$). The critical spanning probability, $R(p_c)$, is also numerically studied, for systems with linear sizes ranging from L=32 up to L=480: the value we found, $R(p_c)=0.270 \pm 0.005$, is the same as for usual percolation with free boundary conditions.