ACTIVATED RANDOM WALKS ON Zd
Palestrante: Leonardo Trivellato Rolla (IME - USP)
Duração: 3 aulas
Datas: 15 à 19 de Janeiro/2024
Local: Sala B110 do Centro de Tecnologia (CT) - UFRJ
Resumo: Some stochastic systems are particularly interesting as they exhibit critical behavior without fine-tuning of a parameter, a phenomenon called self-organized criticality. In the context of driven-dissipative steady states, one of the main models is that of Activated Random Walks. Long-range effects intrinsic to the conservative dynamics and lack of a simple algebraic structure cause standard tools and techniques to break down. This makes the mathematical study of this model remarkably challenging. Yet, some exciting progress has been made in the last ten years, with the development of a framework of tools and methods which is finally becoming more structured. In this minicourse we present the existing results and reproduce the techniques developed so far.
Palestrante: Michael Högele (Universidad de Los Andes, Bogotá)
Duração: 5 aulas
Datas: 04 à 08 de Dezembro/2023
Local: Sala C116 do Centro de Tecnologia (CT) - UFRJ
Resumo: The cutoff phenomenon is a classical threshold phenomenon for the thermalization of a given stochastic model, such as random walks on finite groups modelling card shuffling, for instance, to its equilibrium along a certain time scale. More precisely, assume a parametrization of a family of processes, its limiting measures, and a family of renormalized distances. The mentioned parameter can be for example the size n of a deck of cards or (in the case of a stochastic differential equation) the noise amplitude epsilon. The cutoff phenomenon establishes a parameter-dependent time scale t such that in the limit of the parameter the distance between the current state is ever sharper divided between large values of the distance, when the system lags behind the time scale t (”to the left”) and small values when the system is ahead of t (”to the right”). It has been studied in many discrete situations (time and space) since the beginning by the seminal works of Aldous and Diaconis. In most settings it is shown in terms of the total variation distance, which in continuous space presents a too strong topology in general since it is discontinuous under discrete approximations, which leads to somewhat artificial smoothing assumptions on noise and the invariant measure in several higher-dimensional settings. In a series of papers, the speaker and co-authors have studied the cutoff phenomenon for linear and nonlinear stochastic (partial) differential equations with small and non-small noise mostly in the Wasserstein distance. This minicourse offers an introduction to the subject and its proof methods.
Os slides do minicurso podem ser acessados: AQUI.