Abelian Cascade Dynamics in Bootstrap Percolation

TL;DR

This paper introduces an efficient algorithm for determining bootstrap percolation thresholds leveraging the Abelian property, and explores phase transitions in generalized and continuum models.

Contribution

It presents a novel $L^2$ time algorithm for bootstrap percolation thresholds and analyzes phase transitions in generalized and continuum models.

Findings

Efficient $L^2$ algorithm for bootstrap percolation threshold determination.

Continuous phase transitions observed in generalized bootstrap percolation.

Similar phase transition results found in continuum bootstrap percolation.

Abstract

The culling process in Bootstrap Percolation is Abelian since the final stable configuration does not depend on the details of the updating procedure. An efficient algorithm is devised using this idea for the determination of the bootstrap percolation threshold in two dimension which takes $L^2$ time compared to the $L^3 \log L$ in the conventional method. A generalised Bootstrap Percolation allowing many particles at a site is studied where continuous phase transitions are observed for all values of the threshold parameter. Similar results are also obtained for the continuum Bootstrap Percolation model.