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 $\ast$-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.

15 pages

References in corpus (2)

• The Percolation Transition in the Zero-Temperature Domany Model
• A Particular Bit of Universality: Scaling Limits of Some Dependent Percolation Models