Scaling limit and critical exponents for two-dimensional bootstrap percolation

Consider a cellular automaton with state space {0, 1}Z² where the initial configuration ω₀ 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 ω_n at time n converges exponentially fast to a final configuration. (Formula Presented.), and that the limiting measure corresponding to (Formula Presented.) is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents β, η, ν and γ, and of the continuum scaling limit of crossing probabilities for independent site percolation on the close-packed version of Z² (i.e. for independent ∗-percolation on Z²), 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.