Metastability effects in bootstrap percolation

Bootstrap percolation models, or equivalently certain types of cellular automata, exhibit interesting finite-volume effects. These are studied here at a rigorous level. We find that for an initial configuration obtained by placing particles independently with probability p«1, at sites of Z^d(d≥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 finaldensity as a function of p changes from 0 to 1 around a value which varies only slowly with L— pertinent parameter beingλ=p^1/(d−1) In L. The driving mechanism is the capture of a ‘critical droplet’. This behaviour is analogous to the decay of a metastable statenear a first-order phase transition, for which the present analysis offers some suggestive ideas.