Seminários de probabilidade – 2020

Coordenação: Professor Guilherme Ost e Professora Maria Eulalia Vares

Devido à pandemia de coronavírus, as palestras ocorrerão no ambiente virtual gratuito do Google Meet (https://meet.google.com/nxh-optr-wtq) durante os próximos meses. As palestras ocorrerão às segundas-feiras às 15h, a menos de algumas exceções devidamente indicadas.

Lista completa

The task of efficiently generating uniform spanning trees of a graph has received much attention. A breakthrough came with Aldous-Broder and Wilson’s algorithms, which can efficiently generate spanning trees based on random walks. In this work, we study the transient behavior of both algorithms. We introduce the notion of branches, which are paths generated by the two algorithms on particular stopping times. This interpretation is used to show a transient equivalence between the two algorithms on complete graphs. This equivalence yields a hybrid approach to generate uniform spanning trees of complete graphs faster than either of the two algorithms. We also propose a two-stage framework to explore this hybrid approach beyond complete graphs, showing its feasibility in some examples.

We consider a stretched version of the square lattice where the distances between neighboring vertical columns are given by interarrival intervals of a renewal process. Hence, horizontal edges that link vertices in the same pair of vertical columns have a common random length while every vertical edge has length one. Conditioned on the realization of the lattice, we define a bond percolation model where edges are open with probabilities that depend on their length. We relate the question of whether the model undergoes a non-trivial phase transition to the moments of interarrival times of the renewal process governing the distance among columns. We will also discuss some other related percolation models defined on media with similar types of columnar disorder. Based on a joint work with Marcos Sá, Augusto Teixeira and Remy Sanchis.
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Importance sampling is a widely used technique to estimate the properties of a distribution. The resulting estimator is always unbiased, but may sometimes incur huge or infinite variance. This work investigates trading-off some bias for variance by winsorizing the importance sampling estimator using an adaptive thresholding procedure based on the Balancing Principle (also known as Lepskii’s Method). This provides a principled way to perform winsorization, with finite-sample optimality guarantees and good empirical performance.

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The objective of the talk is to discuss a long range percolation model on oriented trees which contains, as special cases, models such as the frog model with random lifetime and others we may present if time allows. We will be specially interested in localizing, as precisely as possible, the critical parameters.
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We consider some problems related to the truncation question in long-range percolation. It is given probabilities that certain long-range oriented bonds are open; assuming that these probabilities are not summable, we ask if the probability of percolation is positive when we truncate the graph, disallowing bonds of range above a possibly large but finite threshold. This question is still open if the set of vertices is $Z^2$. We give some conditions in which the answer is affirmative. One of these results generalize the previous result in [Alves, Hilário, de Lima, Valesin, Journ. Stat. Phys. {bf 122}, 972 (2017)]. Joint work with Alberto M. Campos.
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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).
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The KPZ equation is the stochastic partial differential equation in d space dimensions formally given by partial_t h=Delta h +langle h,Q hrangle +xi, where xi is the so called space time white noise, i.e., a gaussian process with short range correlations, and Q is a d dimensional matrix. This equation was introduced in the physics literature in the late eighties to model stochastic growth phenomena, is moreover connected to (d+1) dimensional directed polymers in a random potential and is supposed to arise as a scaling limit of a large class of interacting particle systems.

In this talk I will try to explain where this equation comes from, why it is interesting, and how its behaviour depends on the spatial dimension. I will mostly focus on the case of dimension 2, and I will comment on a recent result which contradicts a folklore belief from the physics literature.
This is based on joint works with Giuseppe Cannizzaro, Philipp Schönbauer and Fabio Toninelli.

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All models in the 1+1 Kardar-Parisi-Zhang (KPZ) universality class have fluctuations that converge under KPZ scaling to a universal Markov process, named the KPZ fixed point. In this talk we consider this universal process starting from a two-sided Brownian motion with an arbitrary diffusion coefficient. We apply the integration by parts formula from Malliavin calculus to establish a key relation between the two-point (correlation) function and the location of the maximum of an Airy process plus a Brownian motion with a negative parabolic drift. Integration by parts also allows us to deduce the density of this location in terms of the second derivative of the variance of the KPZ fixed point. We further develop an adaptation of Malliavin-Stein method that implies asymptotic independence with respect to the initial data.
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We consider a natural random growth process with competition on Z^d called first-passage percolation in a hostile environment, that consists of two first passage percolation processes FPP_1 and FPP_lambda that compete for the occupancy of sites. Initially FPP_1 occupies the origin and spreads through the edges of Z^d at rate 1, while FPP_lambda is initialised at sites called seeds that are distributed according to a product of Bernoulli measures of parameter p. A seed remains dormant until FPP_1 or FPP_lambda attempts to occupy it, after which it spreads through the edges of Z^d at rate lambda. We will discuss the results known for this model and present a recent proof that the two types can coexist (concurrently produce an infinite cluster) on Z^d. We remark that, though counterintuitive, the above model is not monotone in the sense that adding a seed of FPP_lambda could favor FPP_1.

A central contribution of our work is the development of a novel multi-scale analysis to analyze this model, which we call a multi-scale analysis with non-equilibrium feedback and which we believe could help analyze other models with non-equilibrium dynamics and lack of monotonicity. A crucial step in our analysis is the addition of some non-local events to the multi-scale framework, and interplaying the non-local events with a by now “standard” multi-scale renormalization.

Based on a joint work with Tom Finn (Univ. of Bath).

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Daniel Takahashi (UFRN)

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.

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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.
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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.
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The quantification problem consists of determining the prevalence of a given label in a target population using labels from a sample from the training population. A common assumption in this situation is that of prior probability shift, that is, once the labels are known, the distribution of the features is the same in the training and target populations. In this paper, we derive a new lower bound for the risk of the quantification problem under the prior shift assumption. Complementing this lower bound, we present a new approximately minimax class of estimators, ratio estimators, which generalize several previous proposals in the literature. Using a weaker version of the prior shift assumption, which can be tested, we show that ratio estimators can be used to build confidence intervals for the quantification problem. We also extend the ratio estimator so that it can:(i) incorporate labels from the target population, when they are available and (ii) estimate how the prevalence of positive labels varies according to a function of certain covariates.
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We consider the parabolic Anderson model in d-dimensional space, i.e., the stochastic heat equation with multiplicative potential, with a random attractive potential having inverse-square singularities on the points of a standard Poisson point process. We study existence and large-time asymptotics of positive solutions via Feynman-Kac representation.
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A family of independent random variables can be associated to the sequence p(n), which counts the number of partitions of a natural number n. The sum of those variables, suitably normalized, can be seen to converge to a Gaussian random variable, which suggests a method to obtain detailed asymptotics for p(n) an n goes to infinity. Moreover, the representation is useful to deduce asymptotic properties when the uniform distribution is considered on the set of partitions of n. The problem is related with questions arising in several contexts.