Título: Anisotropic bootstrap percolation
Palestrante: Daniel Ricardo B. Tordecilla
Data: 1 de abril de 2019 (segunda-feira)
Horário: 15h30
Local: Sala B106-a (Bloco B - CT), Instituto de Matemática - UFRJ
Resumo: Bootstrap percolation is a monotone version of the Glauber dynamics of the Ising model of ferromagnetism. In this talk we will consider nisotropic bootstrap models, which are three-dimensional analogues of a family of (two-dimensional) processes studied by Duminil-Copin, van Enter and Hulshof. In these models the underlying graph G has vertex set [L]3, and the neighbourhood of each vertex consists of the ai nearest neighbours in the ei-direction for each i ∈ {1, 2, 3}, where a1 ≤ a2 ≤ a3. Given an initial configuration in {0, 1}V(G), the system evolves in discrete time in the following way: the state of a vertex v changes from 0 to 1 when it has at least r neighbours in state 1. The initial state is usually chosen to be the product of Bernoulli easures with density p, and the main question is to determine the so-called critical length for percolation Lc(p), for small values of p.
It turns out that Lc(p) is polynomial if r ≤ a3, exponential if a3 < r ≤ a2 + a3, doubly exponential if a2 + a3 <r ≤ a1 + a2 + a3, and infinite if r > a1 + a2 + a3. In this talk we will focus on the case r = a3 + 1, and show how to determine log Lc(p) up to a constant factor. The main new tool, which we call the beams process, allows one to reduce the problem to proving an exponential decay property for a certain two-dimensional model whose behaviour resembles site percolation.
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