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Palestra: Spanning subgraphs of random graphs
Palestrante: Rob Morris (IMPA)

Data: 29/04/2019 (segunda-feira)
Horário: 15:30h
Local: Sala B106-a (Bloco B - CT), Instituto de Matemática - UFRJ

Resumo: Let H be a graph on n vertices with maximum degree at most d. What is the threshold for the appearance of H in the Erdos-Renyi random graph G(n,p)? A well-known conjecture states that for every such H the threshold is at most n^{-2/(d+1)} (log n)^{O(1)}, and this has been proved for "nowhere dense" graphs by Riordan, and for graphs with bounded components by Johansson, Kahn and Vu. In this talk we will discuss some recent progress on this conjecture for odd values of d.

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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