Seminários de Probabilidade 2022 - 1º Semestre

11/07

In the talk we will consider a problem of small ball probabilities for Gaussian processes, which consists in finding the asymptotics of probability that a norm of a process is less than “epsilon” as “epsilon” tends to zero. This question arises in different areas: quantization of Gaussian vectors, metric entropy, etc. We will consider what is already known in the general situation and talk about more advanced results in L_2-norm, for which the distribution is totally defined by eigenvalues of the covariance operator.

04/07
20/06
06/06
23/05

Exponential Random Graphs are defined through probabilistic ensembles with one or more adjustable parameters. They can be seen as a generalization of the classical Erdos Renyi random graph, obtained by defining a tilted probability measure that is proportional to the densities of certain given finite subgraphs. In this talk we will focus on the edge-triangle model, that is a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. Borrowing tools from statistical mechanics, together with large deviations techniques, we will characterize the limiting behavior of the edge density for all parameters in the so-called replica symmetric regime, where a complete characterization of the phase diagram of the model is accessible. First, we determine the asymptotic distribution of this quantity, as the graph size tends to infinity, in the various phases. Then we study the fluctuations of the edge density around its average value off the critical curve and formulate conjectures about the behavior at criticality based on the analysis of a mean-field approximation of the model.

Joint work with Francesca Collet and Elena Magnanini (available at https://arxiv.org/abs/2105.06312).

16/05 (excepcionalmente às 15h30 e exclusivamente em formato presencial na sala B106B, Centro de Tecnologia)

In this talk, we consider a particle moving linearly inside generalized parabolic domain with Markovian reflection at the boundary. Our goal is to determine, with respect to the reflection law, when the process is recurrent or transient. The central idea, which applies a class of non-homogeneous random walks, is to find a Lyapunov function for the properly rescaled version of the problem. Next, we employ supermartingale methods to identify, given the reflection kernel, a phase transition, from recurrent to transient, as we change a single geometric parameter of the region for these stochastic billiards. The solution of this problem is done via a translation to a broad class of almost Markovian models, the half-strip models, and makes use of some tools from functional analysis.

This is a joint work with Mikhail Menshikov and Andrew Wade.

Access the slides here.

09/05

We study an extension of the generalized excited random walk (GERW) on $mathbb{Z}^d$ introduced in [{em Ann. Probab. 40 (5), 2012}}] by Menshikov, Popov, Ramírez and Vachkovskaia. Our extension consists in studying a version of the GERW where excitation may/may not occur according to a time-dependent probability. Specifically, given a sequence of parameters ${p_n}_{n ge 1}$, with $p_n in (0, 1]$ for all $n ge 1$, whenever the process visits a site at time $n$ for the first time, with probability $p_n$ it gains a drift in a given direction (could be any direction of the unit sphere). Otherwise, with probability $1-p_n$, it behaves as a $d$-martingale with zero-mean vector. Whenever the process visits an already-visited site, the process acts again as a $d$-martingale with zero-mean vector. We refer to the model as a GERW in Bernoulli environment, in short $p_n$-GERW.

Under the same hypothesis of [{em Ann. Probab. 40 (5), 2012}}] (bounded jumps, uniform ellipticity) and with a sequence ${p_n}_{n ge 1}$ which decays polynomially, namelly $p_n = mathcal{C}n^{-beta} wedge 1$ with $beta > 0$ and $mathcal{C}$ is a positive constant, we show a series of results for the $p_n$-GERW depending on the value of $beta$ and on the dimension. Specifically, for $beta < 1/6$ and $dgeq 2$, we show that the $p_n$-GERW has a positive probability of never returning to the origin in the drift direction, for $beta > 1/2$, $dgeq 2$ and $beta=1/2$ and $d=2$ we obtain, under certain conditions, a Functional Central Limit Theorem. Finally, for $beta=1/2$ and $d ge 4$ we obtain, under suitable conditions, that the $p_n$-Name{} is a tight process, and every limit point $Y$ satisfies $W_t cdot ell + c_1 sqrt{t} preceq Y_t cdot ell preceq W_t cdot ell + c_2 sqrt{t}$ where $c_1$ and $c_2$ are positive constants, $W$ is a Brownian motion and $ell$ is the direction of the drift.

Access the slides here.

Assista à palestra no Youtube
25/04

We consider general classes of integer-valued gradient models, including the SOS-model, on infinite regular trees. We discuss constructive results on the existence of localized Gibbs measures and delocalized gradient Gibbs measures, both in regimes of strong coupling. We also discuss the existence of measures which are not invariant under translations on the tree.

Joint work with Florian Henning.

Access the slides here.

Assista à palestra no Youtube
11/04

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).

Access the slides here.

Assista à palestra no Youtube
07/03

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.

This is a joint work with Daniel Y. Takahashi (Ice/UFRN).

Access the slides here.

Assista à palestra no Youtube
21/02

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.

Access the slides here.

Assista à palestra no Youtube
07/02

In most of the theoretical research about random systems with local interaction, the supposition that the set of particles, also called the space, does not change during the evolution is widely adopted. This supposition, which we call constant length, is not the only possible one. We will consider another approximation, called variable length. In this talk, for a class of one-dimensional particle processes with discrete-time, where its components are located in the integers set, we shall present a general theory and analyze (in a theoretical and numerical way) some random processes, studied by me with co-authors, which were motivated by this new paradigm.

Access the slides here.

Assista à palestra no Youtube
24/01

I will sketch a proof of the following result: If a positive-rate probabilistic cellular automaton has a Bernoulli invariant measure, then it is ergodic, meaning that its distribution starting from any configuration converges (exponentially fast) to that Bernoulli measure. The same result holds for (continuous-time) interacting particle systems. The proof is based on the entropy method, but unlike the usual entropy arguments, does not require the shift-invariance of the starting measure. This result has a practical implication concerning the inevitability of heat dissipation in computers which I will briefly discuss. This is a joint work with Irène Marcovici.

Access the slides here.

Assista à palestra no Youtube
10/01 (ADIADO)

We develop a novel cluster expansion for finite-spin lattice systems subject to multi-body quantum — and, in particular, classical — interactions. Our approach is based on the use of “decoupling parameters”, advocated by Park, which relates partition functions with successive additional interaction terms. Our treatment, however, leads to an explicit expansion in a beta-dependent effective fugacity that permits an explicit evaluation of free energy and correlation functions at small beta. To determine its convergence region we adopt a relatively recent cluster summation scheme that replaces the traditional use of Kikwood-Salzburg-like integral equations by more precise sums in terms of particular tree-diagrams.

Joint work with Nguyen Tong Xuan.