Seminários de probabilidade – Segundo Semestre de 2024

Quando forem online, as palestras ocorrerão via Google Meet às segundas-feiras às 15h30, a menos de algumas exceções devidamente indicadas.

Quando forem presenciais, as palestras ocorrerão na sala C-116 às segundas-feiras às 15h30, a menos de algumas exceções devidamente indicadas.

Todas as palestras são em inglês.

Lista completa (palestras futuras podem sofrer alterações)

In this talk, we will provide a brief history of some recent developments in the study of random walks in dynamic random environments. Then, we will present a new result that establishes the Central Limit Theorem (CLT) for a family of environments that mix rapidly but not uniformly. We will give an outline of the proof, which follows a very simple idea: showing that the random walk cannot escape from a set of renewal traps. We will conclude the seminar by indicating some interesting directions for the continuation of this study.

This presentation is based on joint work with Julien Allasia, Rangel Baldasso, and Oriane Blondel

In this talk we will discuss some recent developments in random walks in dynamic random environments, when the environment is given by a realization of a particle system as the SEP or ZRP.

We use the framework of Quantitative Hydrodynamics to derive a CLT around its hydrodynamic limit for an interacting particle system inspired by integrate-and-fire neuron models. The hydrodynamic limit of this model was originally derived by Chariker, De Masi, Lebowitz and Presutti, and as an important intermediate step we show that this convergence holds at optimal L^2-speed.

Joint with Julian Amorim and Yangrui Xiang.

I will present some recent results which have been obtained for the facilitated exclusion process, in one dimension. This stochastic lattice gas is subject to strong kinetic constraints which create a continuous phase transition to an absorbing state at a critical value of the particle density. If the microscopic dynamics is symmetric, its macroscopic behavior, under periodic boundary conditions and diffusive time scaling, is ruled by a non-linear PDE belonging to free boundary problems (or Stefan problems). One of the ingredients is to show that the system typically reaches an ergodic component in subdiffusive time. When the particle system is put in contact with reservoirs of particles (which can either destroy or inject particles at both boundaries), we observe an usual impact on the boundary values of the empirical density. Based on joint works with O. Blondel, H. Da Cunha, C. Erignoux, M. Sasada and L. Zhao.

Given a finite transitive graph and n particles labeled with numbers {1,2,3,…,n}, we place these particles on the set of vertices at random. Then, we let the particles evolve as a system of coalescing random walks: each particle performs a continuous-time simple random walk (SRW) and whenever two particles meet, they merge into one particle which continues to perform a SRW. At each time t, consider the partition P_t of {1,2,3,…,n} induced by the equivalence relation: i~j when particles i and j occupy the same vertex at time t. We show that the Kingman n-coalescent model emerges as a scaling limit for (P_t), as n is fixed and the size of the graph goes to infinity. This result allows me to talk about our main tool in our approach: the martingale problem.

Joint work with Enrique Chavez.

 

 

The characterization of the invariant measures for non-reversible particle systems driven out-of-equilibrium via the action of external reservoirs is typically a difficult task. This has been achieved e.g. for the well-known exclusion process. In this talk I will show a class of boundary driven zero-range models whose non-equilibrium steady state can be explicitly characterized via a probabilistic mixture of inhomogeneous product measures. This characterization of the non-equilibrium steady state allows to compute the formula for the density large deviation function predicted by Macroscopic Fluctuation Theory and to establish the additivity principle.

This is from joint works with: Chiara Franceschini, Rouven Frassek, Davide Gabrielli, Cristian Giardinà, Frank Redig and Dimitrios Tsagkarogiannis

Link Youtube

 

We study Activated Random Walks on a unidimensional ring \mathbb{Z}/N \mathbb{Z}. In this conservative model, a number \zeta* N of particles is distributed on the unidimensional ring. Those particles are either active or sleeping. Active particles perform independent continuous-time random walks at rate 1 and fall asleep at rate \lambda. Sleeping particles do not move. If \zeta<1, this system will eventually be attracted to some configuration with only sleeping particles. We will prove that, for every sleeping rate \lambda, activity will be sustained for a long time (exponential in the size of the block) if the density \zeta is close enough to 1.