Metastability effects in bootstrap percolation

Abstract

Bootstrap percolation models, or equivalently certain types of cellular automata, exhibit interesting finite-volume effects. These are studied at a rigorous level. The authors find that for an initial configuration obtained by placing particles independently with probability p or=2), the density of the 'bootstrapped' (final) configurations in the sequence of cubes (-L/2, L/2)d typically undergoes an abrupt transition, as L is increased, from being close to 0 to the value 1. With L fixed at a large value, the mean final density as a function of p changes from 0 to 1 around a value which varies only slowly with L-the pertinent parameter being lambda =p1(d-1)/ln L. The driving mechanism is the capture of a 'critical droplet'. This behaviour is analogous to the decay of a metastable state near a first-order phase transition, for which the analysis offers some suggestive ideas.