Clarification of the Bootstrap Percolation Paradox

TL;DR

This paper introduces a new simulation method and theoretical framework to better understand the bootstrap percolation transition, allowing for more accurate comparisons between theory and simulation at higher densities.

Contribution

It develops a novel importance-sampling simulation technique and a systematic theoretical correction for the bootstrap percolation model, improving analysis of the transition.

Findings

Enhanced simulation access to larger bootstrap lengths.

Systematic corrections improve theoretical predictions.

Credible theory-simulation comparisons achieved.

Abstract

We study the onset of the bootstrap percolation transition as a model of generalized dynamical arrest. We develop a new importance-sampling procedure in simulation, based on rare events around "holes", that enables us to access bootstrap lengths beyond those previously studied. By framing a new theory in terms of paths or processes that lead to emptying of the lattice we are able to develop systematic corrections to the existing theory, and compare them to simulations. Thereby, for the first time in the literature, it is possible to obtain credible comparisons between theory and simulation in the accessible density range.