**Palestrante:** Rodrigo Lambert (UFU)**Data:** 22/02/2021**Horário:** 15h - 16h (Rio de Janeiro local time)**Local:** Transmissão online

Clique **AQUI** para acessar a transmissão.

**Resumo:** For a string of length n, the overlapping function defines the greatest size of a repetition, in the sense that it is k if its first and last k symbols coincide. When the source that generates the strings satisfies the complete grammar condition, the overlapping function is always non-negative. In the present work we deal with the case where the complete grammar condition is removed, and therefore “negative overlaps” are allowed. We state a weak convergence theorem when the source is a beta-mixing Markov Chain with finite diameter (greatest “distance” between two symbols of the alphabet). This is a work in progress in collaboration with Erika Alejandra Rada-Mora (UFABC).

Todas as palestras são realizadas em inglês.

Aproveitamos para informar que os vídeos dos seminários online realizados durante 2020 estão disponíveis **AQUI**.

Em relação a este ano, alguns dias após cada encontro o vídeo deverá estar disponível **AQUI**.

**Título****:** Conditional propagation of chaos for systems of interacting neurons with random synaptic weights

**Palestrante****:** Eva Löcherbach (Université Paris I)**Data:** 08/02/2021**Horário****:** 15:00 até 16:00h (Horário do Rio de Janeiro)**Local****:** Transmissão on-line

Confira **AQUI**** **o link para a transmissão.

**Resumo****:** We study the stochastic system of interacting neurons introduced in De Masi et al 2015, in a diffusive scaling. The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to 0 and all other neurons receive an additional amount of potential which is a centred random variable of order 1 / \sqrt (N). In between successive spikes, each neuron's potential follows a deterministic flow. We prove the convergence of the system, as the number of neurons tends to infinity, to a limit nonlinear jumping stochastic differential equation driven by Poisson random measure and an additional Brownian motion W which is created by the central limit theorem. This Brownian motion is underlying each particle's motion and induces a common noise factor for all neurons in the limit system.

Conditionally on W, the different neurons are independent in the limit system. This is the ``conditional propagation of chaos'' property. We prove the well-posedness of the limit equation by adapting the ideas of Graham 1992 to our frame. To prove the convergence in distribution of the finite system to the limit system, we introduce a new martingale problem that is well suited for our framework. The uniqueness of the limit is deduced from the exchangeability of the underlying system.

This is a joint work with Xavier Erny and Dasha Loukianova, both of university of Evry.

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

Clique **AQUI** para acessar a transmissão.

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**: Central limit theorems for a driven particle in a random medium with mass aggregation

**Palestrante**: Pablo Almeida Gomes (IME-USP)**Data: **01/02/2021**Horario**: 15:00 até 16:00h (Horário do Rio de Janeiro)**Local**: Transmissão online

Confira **AQUI** o link para a transmissão.

**Resumo**: In this talk we investigate a one-dimensional infinite mechanical particle system, driven by a constant force F. The system consists of one charged particle, together with field-neutral ones. Neutral particles are initially randomly placed in the medium, and can be perfectly elastic or inelastic, according to independent Bernoulli random variables. We establish central limit theorems for the velocity and position of the charged particle.

Based on joint work with Luiz Renato Fontes and Rémy Sanchis.

**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.

Confira **AQUI** o link para a transmissão.

**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.