Local Bootstrap Percolation

TL;DR

This paper analyzes a variant of bootstrap percolation where growth is limited to a single cluster, providing precise probabilistic bounds and correcting earlier numerical predictions about the scaling behavior.

Contribution

It introduces a new model of bootstrap percolation with restricted growth and derives rigorous bounds on the probability of full activation, correcting previous numerical estimates.

Findings

Probability of full activation is exp[alpha(p)/p]

Bounds on alpha(p) involve pi^2/9 and sqrt p corrections

Corrects earlier numerical predictions for the correction term

Abstract

We study a variant of bootstrap percolation in which growth is restricted to a single active cluster. Initially there is a single active site at the origin, while other sites of Z^2 are independently occupied with small probability p, otherwise empty. Subsequently, an empty site becomes active by contact with 2 or more active neighbors, and an occupied site becomes active if it has an active site within distance 2. We prove that the entire lattice becomes active with probability exp[alpha(p)/p], where alpha(p) is between -pi^2/9 + c sqrt p and pi^2/9 + C sqrt p (-log p)^3. This corrects previous numerical predictions for the scaling of the correction term.