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 a_i nearest neighbours in the e_i-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 L_c(p), for small values of p.

It turns out that L_c(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 L_c(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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