Slow Convergence in Bootstrap Percolation

TL;DR

This paper investigates the slow convergence of the bootstrap percolation phase transition, providing rigorous bounds that explain discrepancies between simulations and theoretical asymptotics for large system sizes.

Contribution

It establishes explicit bounds on the convergence rate of the critical parameter in bootstrap percolation, highlighting the slow approach to the asymptotic limit.

Findings

Discrepancy between critical parameter and its limit is at least Omega((log L)^(-1/2))

Critical window width is Theta((log L)^(-1))

Relative discrepancy can be at least 1% even for L=10^3000

Abstract

In the bootstrap percolation model, sites in an L by L square are initially infected independently with probability p. At subsequent steps, a healthy site becomes infected if it has at least 2 infected neighbours. As (L,p)->(infinity,0), the probability that the entire square is eventually infected is known to undergo a phase transition in the parameter p log L, occurring asymptotically at lambda = pi^2/18. We prove that the discrepancy between the critical parameter and its limit lambda is at least Omega((log L)^(-1/2)). In contrast, the critical window has width only Theta((log L)^(-1)). For the so-called modified model, we prove rigorous explicit bounds which imply for example that the relative discrepancy is at least 1% even when L = 10^3000. Our results shed some light on the observed differences between simulations and rigorous asymptotics.