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Palestra: Stationary states of symmetric exclusion processes with complex boundary dynamics
Palestrante: Tiecheng Xu (IMPA)

Data: 15 de abril de 2019 (segunda-feira)
Horário: 15:30h
Local: Sala B106-a (Bloco B - CT), Instituto de Matemática - UFRJ

Resumo: In this talk we will discuss the stationary sates of the one-dimensional, boundary driven, symmetric exclusion processes with some non-reversible boundary dynamics. We mainly focus on the exclusion processes whose boundary dynamics are the small perturbation of flipping dynamics. I am going to explain how to derive the hydrostatic limit of this type of processes using duality techniques. If time permits, I will also mention the results for the processes with some other types of boundary dynamics. Joint work with C. Erignoux and C. Landim.

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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