Cellular Automata with Rare Events; Resolution of an Outstanding Problem in the Bootstrap Percolation Model

摘要

We study the modified bootstrap model as an example of a cellular automaton that is dominated by the physics of rare events. For this reason, the characteristic laws are quite different from the simple power laws often found in well known critical phenomena. Those laws are quite delicate, and are driven by co-operative events at diverging length and time scales. To deal with such problems we introduce a new importance-sampling procedure in simulation, based on rare events around "holes". This enables us to access bootstrap lengths beyond those previously studied. By studying the paths or processes that lead to emptying of the lattice we are able to develop systematic corrections to the 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.