Título: An Effective Class of Ballistic Random Walks in Mixing Random Environments

Palestrante: Glauco Valle (IM-UFRJ)
Data: 25/01/2021
Horário: 15:00 -  16:00. (Rio de Janeiro local time)
Local: Transmissão online

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Resumo: We study d-dimensional random walks in strong mixing environments (RWRE), with underlying dimension d>=2. Under a suitable polynomial effective condition, we prove a functional central limit theorem of ballistic type. Specifically, we construct a new effective criterion equivalent to usual ballisticity conditions. This construction allows us to prove, in a mixing framework, the RWRE conjecture regarding the equivalence between ballisticity conditions already proved for iid environments. We then obtain the polynomial effective condition that provides the existence of arbitrary finite moments for approximate regeneration times, yielding the central limit theorem for the RWRE.

Joint work with Maria Eulalia Vares (UFRJ) and Enrique Guerra (PUC-Chile).

All the talks are held in English.

We take the opportunity to inform that the videos of the online seminars held during 2020 are available HERE.

Regarding this year, a few days after each meeting the video should be available HERE.

Título: Brownian modules of continuité and diffusion approximation

Palestrante: Julien Chevallier (LJK, Université Grenoble Alpes)
Data: 11/01/2021
Horário: 15:00h - 16:00h (Rio de Janeiro local time)
Local: Transmissão Online

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Resumo: Garsia–Rodemich–Rumsey (1971) proved an inequality which has been used to upper-bound the Brownian modules of continuity. In turn, this upper-bound was used by T.G. Kurtz (1976) to prove a strong diffusion approximation result for pure jump processes. However, this proof makes the crucial assumption that jump rates are uniformly bounded. The main objective of this talk is to show how to get rid of this assumption starting back from GRR inequality. The second objective is to show that this scheme of proof is robust to time scaling.

All the talks are held in English.
We take the opportunity to inform that the videos of the online seminars held during 2020 are available HERE.

Regarding this year, a few days after each meeting the video should be available HERE.

Título: Non-intersecting Brownian motions with outliers, KPZ fluctuations and random matrices

Palestrante: Daniel Remenik (Universidad de Chile)
Data: 18/01/2020
Horario: 15:00 - 16:00 (Horário do Rio de Janeiro)
Local: Transmissão online.

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Resumo: A well known result implies that the rescaled maximal height of a system of N non-intersecting Brownian bridges starting and ending at the origin converges, as N goes to infinity, to the Tracy-Widom GOE random variable from random matrix theory. In this talk I will focus on the same question in case where the top m paths start and end at arbitrary locations. I will present several related results about the distribution of the limiting maximal height for this system, which provides a deformation of the Tracy-Widom GOE distribution: it can be expressed through a Fredholm determinant formula and in terms of Painlevé transcendents; it corresponds to the asymptotic fluctuations of models in the KPZ universality class with a particular initial condition; and it is connected with two PDEs, the KdV equation and an equation derived by Bloemendal and Virag for spiked random matrices. Based on joint work with Karl Liechty and Gia Bao Nguyen.

Título: Renewal Contact Process: phase transition and survival

Palestrante: Daniel Ungaretti (IME-USP)
Data: 04/01/2021
Horario: 15:00 - 16:00 (Horário do Rio de Janeiro)
Local: Transmissão Online

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Resumo: The Contact Process was introduced by Harris in 1974 and models the spread of an infection on a graph. The state of each vertex is either infected or healthy, and there are two competing factors that govern the evolution of the process over time: infected vertices become healthy at rate 1 and healthy vertices can get infected at a rate proportional to its current number of infected neighbors. In two recent papers, Fontes, Marchetti, Mountford and Vares introduced a generalization of the model in which cures are given by renewal processes with some fixed interarrival distribution. I will discuss how the choice of interarrival distribution affects the spread of the infection, focusing on recent developments in which we improved the characterization of the interarrival distributions for which there is phase transition. Joint work with Luiz Renato Fontes, Tom Mountford and Maria Eulália Vares.