Titulo: Asymptotic behavior of a low temperature non-cascading 2-GREM dynamics at extreme time scales

Palestrante: Leonel Zuaznábar (IME-USP)

Data: 20/06/2022
Horário: 15:00h às 16:00h (Rio de Janeiro local time)
Local: Transmissão online

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Resumo: We are interested in the study of a class of spin glass models introduced by Derrida under the name of Generalized Random Energy Models (GREM). They are based on Gaussian random variables on the hypercube {-1,1}^N with an inherent hierarchical correlation structure. More specifically, we consider the 2-GREM model evolving under the Random Hopping Dynamics at extreme time scales, where it is close to equilibrium, and visits the configurations in the support of the Gibbs measure. There are two scenarios that may be distinguished: the cascading case (studied by Luiz Renato Fontes and Veronique Gayrard in 2019) and the non-cascading case. The cascading occurs when the ground states energies are achieved by adding up the ground states energies of the two levels. In the non-cascading case the correlations are too weak to have an impact on the extremes and the system "collapses'' to a Random Energy Models (REM). Also, in this case, the extremal configurations must differ in the first index, this phenomenon justifies the non-cascading denomination for our system. In this work we got some results about the asymptotic behavior of the GREM model and the Random Hopping Dynamics in the non-cascading case.

Joint work with Luiz Renato Fontes and Susana Frómeta.

Organizadores: Giulio Iacobelli and Maria Eulalia Vares

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Título: Sparse Markov Models for High-dimensional Inference

Palestrante: Daniel Y. Takahashi (Ice/UFRN)
Data: 07/03/2022
Horário: 15:00h
Local: Transmissão online

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Resumo: Finite order Markov models are theoretically well-studied models for dependent categorical data. Despite their generality, application in empirical work when the order is larger than one is quite rare. Practitioners avoid using higher order Markov models because (1) the number of parameters grows exponentially with the order, (2) the interpretation is often difficult. Mixture of transition distribution models (MTD) were introduced to overcome both limitations. MTD represent higher order Markov models as a convex mixture of single step Markov chains, reducing the number of parameters and increasing the interpretability. Nevertheless, in practice, estimation of MTD models with large orders is still limited because of the curse of dimensionality and high algorithm complexity. Here, we prove that if only few lags are relevant we can consistently and efficiently recover the lags and estimate the transition probabilities of high order MTD models. The key innovation is a recursive procedure for the selection of the relevant lags of the model. Our results are based on (1) a new structural result of the MTD and (2) an improved martingale concentration inequality. Our theoretical results are illustrated through simulations.

All the talks are held in English.

More complete information about the seminars will be available at HERE.

Título: Relative entropy and sharp convergence to stationarity of the exclusion process

Palestrante : Milton Jara (IMPA)
Data : 11/04/2022
Horário : 15:00h (Rio de Janeiro local time)
Local : Transmissão online

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Resumo: After comprehensive work by Lacoin et al., the convergence to equilibrium of the one-dimensional exclusion process is well understood. In particular, the mixing time and cut-off window are explicitly known. For the particular example of the reversible exclusion process in contact with reservoirs, we show how the mixing time depends on initial conditions which are relevant in the hydrodynamic limit of the exclusion process. The proof relies on log-Sobolev inequalities, relative entropy methods and the CLT for the density of particles. Joint work with Patricia Gonçalves (Lisbon), Rodrigo Marinho (Porto Alegre) and Otávio Menezes (West Lafayette).

All the talks are held in English. 

More complete information about the seminars will be available at HERE.

Título: On random walks that build their own domain

Palestrante: Rodrigo Ribeiro (University of Colorado Boulder)
Data: 21/02/2022
Horário: 15:00h
Local: Transmissão online

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Resumo: In this talk, we will discuss the dichotomy recurrence/transience and ballisticity in the context of a class of random walks that build their own domain. At each step of the walker on its domain, a random number of new vertices is attached to the walker's position. We will present distributional conditions over this random number of new vertices for which we observe distinct sharp behavior on the walker. We also discuss structural results, that is, when this process is seen as a random graph model, the walker is capable of generating trees whose degree distribution approaches a power-law.

All the talks are held in English.

More complete information about the seminars will be available at HERE.

Sincerely,

Organizers: Giulio Iacobelli and Maria Eulalia Vares