Scaling Limit and Critical Exponents for Two-Dimensional Bootstrap Percolation

Abstract

Consider a cellular automaton with state space $\{0,1 \}^{{\mathbb Z}^2}$ where the initial configuration $\omega_0$ is chosen according to a Bernoulli product measure, 1's are stable, and 0's become 1's if they are surrounded by at least three neighboring 1's. In this paper we show that the configuration $\omega_n$ at time n converges exponentially fast to a final configuration $\bar\omega$, and that the limiting measure corresponding to $\bar\omega$ is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents $\beta$, $\eta$, $\nu$ and $\gamma$, and of the continuum scaling limit of crossing probabilities for independent site percolation on the close-packed version of ${\mathbb Z}^2$ (i.e., for independent $*$-percolation on ${\mathbb Z}^2$), we prove that the bootstrapped percolation model has the same scaling limit and critical exponents. This type of bootstrap percolation can be seen as a paradigm for a class of cellular automata whose evolution is given, at each time step, by a monotonic and nonessential enhancement.