Monte Carlo renormalisation group study of bootstrap percolation

Abstract

The authors develop a Monte Carlo position space renormalisation group method to study the scaling properties of bootstrap percolation. In this model sites on a lattice are randomly occupied with probability p and sites with less than m occupied neighbours are culled until a stable configuration remains. Using progressively larger length rescaling factors b, they obtain numerical results on the triangular lattice for the percolation threshold pc(b), the connectedness length exponent nu (b) and the fractal dimension df(b). The large b limit yields numerical values of df for m=2 and 3 which are consistent with df for ordinary percolation (m=0). Although the small b dependence of nu for m=2 and 3 is qualitatively different from that of ordinary percolation, the large b behaviour of nu suggests that m=2 and m=3 bootstrap percolation on the triangular lattice are in the same universality class as ordinary percolation.