Título: Self-Switching Markov Chains: emerging dominance phenomena.

Palestrante: Guilherme Ost (IM-UFRJ).
Data: 26/10/2020.
Horário: 15:00 to 16:00 (Rio de Janeiro local time)
Local: Transmissão online.

Confira AQUI o link para a transmissão.

Resumo: In many dynamical systems in nature, the law of the dynamics changes along with the temporal evolution of the system. These changes are often associated with the occurrence of certain events. The timing of occurrence of these events depends, in turn, on the trajectory of the dynamical system itself, making the dynamics of the system and the timing of changes in the dynamics strongly coupled. Naturally, trajectories that take longer to satisfy the event will last longer. Therefore, we expect to observe more frequently the dominant dynamics, the ones that take longer to change in the long run. In this talk, we will present a Markov chain model, called Self-Switching Markov Chain (SSMC), in which the emergence of dominant dynamics can be rigorously addressed. We will discuss conditions and scaling in the SSMC under which we observe with probability one only the subset of dominant dynamics. Moreover, we characterize these dominant dynamics. Furthermore, we show that the switching between dynamics exhibits metastability-like property. This is a joint work with Daniel Takahashi (UFRN), Giulio Iacobelli (UFRJ) and Sandro Gallo (UFSCar).

Title: Spatial Gibbs Random Graphs

Speaker: Andressa Cerqueira  (UFSCar)

Our next online seminar will be held next Monday, August 10, from 3 p.m. to 4 p.m. (Rio de Janeiro local time)

The GoogleMeet link for the seminars is HERE.

Abstract:  In this talk, I will present a Spatial Gibbs Random Graph Model on Z^2 that incorporates the interplay between the statistics of the graph and the underlying space where the vertices are located. For this model, we prove the existence and uniqueness of a measure defined on graphs with vertices in Z^2 as the limit along the measures over graphs with finite vertex set. I will explain how the results are obtained based on a graphical construction of the model as the invariant measure of a birth and death process. This is a joint work with Nancy Garcia. 

Title: Mixing rates for processes with long-memory

Speaker: Daniel Yasumasa Takahashi (UFRN)

Our next online seminar will be held next Monday, August 24, from 3 p.m. to 4 p.m. (Rio de Janeiro local time)

The GoogleMeet link for the seminars is HERE. 

Abstract:  Non-Markovian processes are ubiquitous, but they are much less understood compared to Markov processes.  We model non-Markovianity using probability kernels that can depend on its entire history. The continuity rate characterizes how the dependence of kernel on the past decays. One key question is to understand how the mixing rates and decay of correlation are related to the continuity rate. Pollicot (2000) and Bressaud, Fernandez, Galves (1999) showed that if the continuity rate decays as O(1/n^c), for c > 1, then the correlation also decays as O(1/n^c). Johansson, Oberg, Pollicott (2007) proved the uniqueness of the stationary measure compatible with kernels with the continuity rate in O(1/n^c), for c > 1/2. Moreover, Berger, Hoffman, Sidoravicius (2018) established that there are kennels with multiple compatible measures whenever c < 1/2. Therefore, the natural question is to understand the mixing rates and correlation decays when c is in [1/2,1]. In this talk, I will exhibit upper bounds for the mixing rates and correlation decays when the continuity rate decays as  O(1/n^c), for c in (1/2,1].  If time allows, I will show how to apply the result to prove a new weak invariance principle. This talk is based on joint work with Christophe Gallesco.

Título: On the threshold of spread-out contact process percolation
Palestrante: Daniel Valesin (University of Groningen)

Data: 27 de julho de 2020 (segunda-feira)
Hora: 15:00 h

O seminário será realizado no GoogleMeet. Clique AQUI para acessar. 

Abstract: We study the stationary distribution of the (spread-out) d-dimensional contact process from the point of view of site percolation. In this process, vertices of \Z^d can be healthy (state 0) or infected (state 1). With rate one infected individuals recover, and with rate \lambda they transmit the infection to some other vertex chosen uniformly within a ball of radius R. The classical phase transition result for this process states that there is a critical value \lambda_c(R) such that the process has a non-trivial stationary distribution if and only if \lambda > \lambda_c(R). In configurations sampled from this stationary distribution, we study nearest-neighbor site percolation of the set of infected sites; the associated percolation threshold is denoted \lambda_p(R). We prove that \lambda_p(R) converges to 1/(1-p_c) as R tends to infinity, where p_c is the threshold for Bernoulli site percolation on \Z^d. As a consequence, we prove that \lambda_p(R) > \lambda_c(R) for large enough R, answering an open question of [Liggett, Steif, AIHP, 2006] in the spread-out case. Joint work with Balázs Ráth.

Organizadores: Guilherme Ost e Maria Eulalia Vares