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Journée Lyon-Genève
Date : Monday 9 March 2015 Location : ENS de Lyon, site Monod, Amphi A (4th floor) Contact : gregory.miermont@ens-lyon.fr |
Programme
9h30 Welcome coffee10h15 -- 11h05 Matan Harel: Towards The Localization Phase Transition in Random Geometric Graphs with Too Many Edges
11h15 -- 12h05 Juhan Aru: Exploring the GFF
Lunch (buffet)
14h00 -- 14h50 Erich Baur: Percolation on random recursive trees
15h00 -- 15h50 Hugo Duminil-Copin: Scaling window of Bernoulli percolation on Z^d
Coffee break
16h30 -- 17h20 Jean-Christophe Mourrat : The dynamic phi^4 model in the plane
Abstracts
Juhan Aru (Lyon)
Exploring the GFF
In this talk, we will take a closer look at the Gaussian free field, and in particular at what are called the local sets of the GFF, introduced in a paper by Schramm and Sheffield. We discuss where and why they come into the game, and in what sense they are analogues to stopping times of stochastic processes.
Erich Baur (Lyon et FNS)
Percolation on random recursive trees
We explain recent results about cluster sizes of Bernoulli bond percolation on large recursive trees. Moreover, we look at a destruction process on such trees, where edges are cut one after the other in a random uniform order. Partly based on joint work with Jean Bertoin.
Hugo Duminil-Copin (Genève)
Scaling window of Bernoulli percolation on Z^d
We will discuss the notion of scaling window for Bernoulli percolation on Z^d. While the notion is classical in two dimensions, the systematic study of the size of the window in higher dimension has not been done by now. We will present a few progress in this direction.
Matan Harel (Genève)
Towards The Localization Phase Transition in Random Geometric Graphs with Too Many Edges
Consider the Gilbert random geometric graph G(n, r(n)), given by a connecting two points of a Poisson Point Process of intensity n on the unit torus whenever their distance is smaller than the parameter r(n). This model is conditioned on the rare event that the number of edges observed, |E|, is greater than [1 + delta (n)] times its expectation. We show that, when delta is fixed or vanishing sufficiently slowly in n, there exists a "giant clique" with almost all the excess edges forced into the model by conditioning event. If delta vanishes sufficiently quickly, the largest clique will be, at most, a constant multiple of the usual clique number. Finally, we discuss progress in finding a phase transition function delta_(0)(n), so that when delta is much bigger than delta_(0), the giant clique scenario holds, while delta much smaller than delta_(0) implies no giant clique.
Jean-Christophe Mourrat (Lyon)
The dynamic phi^4 model in the plane
The dynamic phi^4 model is a non-linear stochastic PDE which involves a cubic power of the solution. In dimensions 3 and less, solutions are expected to have the same local regularity as solutions of the linearised equation. As it turns out, the solutions of the linearised equation at a given time locally look like a Gaussian free field. Hence, in dimensions 2 and 3, some renormalisation needs to be performed in order to define the cubic power of the solution. In the (full) plane, I will explain how to do this and show that the stochastic PDE has a well-defined solution for all times. If time permits, I will also discuss why the model is the scaling limit of a near-critical Ising model with long-range interactions. Joint work with Hendrik Weber.
