O Seminário de Probabilidade e Mecânica Estatística se reúne às quartas às 13h,
horário de Brasília (16:00 UTC).
Augusto Teixeira, IMPA.
Title:Phase transition for percolation with axes-aligned defects.
In this talk we will review a model that was first introduced by
Jonasson, Mossel and Peres. Starting with the usual square lattice on
Z^2, entire rows (respectively columns) of edges extending along the
horizontal (respectively vertical) direction are removed independently
at random. On the remaining thinned lattice, Bernoulli bond percolation
is performed, giving rise to a percolation model with infinite range
dependencies under the annealed law. In 2005, Hoffman solved the main
conjecture around this model: proving that this percolation process
indeed undergoes a nontrivial phase transition. In this talk, besides
reviewing this surprisingly challenging problem, we will present a novel
proof, which replaces the dynamic renormalization presented previously
by a static version. This makes the proof easier to follow and to extend
to other models. We finally present some remarks on the sharpness of
Hoffman’s result as well as a list of interesting open problems that we
believe can provide a renewed interest in this family of questions.
This talk is based on a joint work with M. Hilário, M. Sá and R. Sanchis.
Elena Kosygina, City University of New York
Convergence and non-convergence of some self-interacting random walks to Brownian motion perturbed at extrema.
Generalized Ray-Knight theorems for edge local times proved to be a very useful tool for studying the limiting behavior of several classes of self-interacting random walks (SIRWs) on integers. Examples include some reinforced random walks, excited random walks, rotor walks with defects. I shall describe two classes of SIRWs introduced and studied by Balint Toth (1996), asymptotically free and polynomially self-repelling SIRWs, and discuss new results which resolve an open question posed in Toth’s paper. We show that in the asymptotically free case the rescaled SIRWs converge to a perturbed Brownian motion (conjectured by Toth) while in the polynomially self-repelling case the convergence to the conjectured process fails in spite of the fact that generalized Ray-Knight theorems clearly identify the unique candidate in the class of all perturbed Brownian motions. This negative result was somewhat unexpected. The question whether there is convergence in the polynomially self-repelling case and, if yes, then how to describe the limiting process is open. This is joint work with Thomas Mountford, EPFL, and Jonathon Peterson, Purdue University.
Eric Luçon, Université Paris Cité.
How large is the mean-field framework ?
The canonical framework for mean-field systems is to consider N
particles (diffusions, or dynamics with jumps etc) that interact on
the complete graph in a uniform way, the strength of interaction
between two particles being of size 1/N. The behavior of the system is
hence captured by the empirical measure of the system which converges
as $N\to\infty$ to the solution of a nonlinear Fokker Planck equation.
The motivation of this talk is simple: what can we say if one no
longer interacts on the complete graph, i.e. one removes connections
between particles ? if the graph is sufficiently close to the complete
graph, one expects the same asymptotic behavior. We will address this
question at the level of the LLN and fluctuations of the empirical
measure of the system.
This is based on joint works with G. Giacomin, S. Delattre, F. Coppini
and C. Poquet.
Alessandra Occelli, Université d'Angers.
Jinho Baik, University of Michigan
Linjie Zhao, Wuhan University.
Timo Seppalainen, University of Wisconsin
Sylvie Méléard, École Polytechnique
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