Palestrantes

Large deviations for inhomogeneous birth-death processes

Anatoly Iambartsev (USP)

Resumo: Birth-death processes form a natural class where ideas and results on Large deviations can be tested. In the talk I overview our (the joint work with A.V. Logachov, Y.M. Suhov, and N.D. Vvedenskaya) results about large deviations for birth-death processes where the jump rate has an asymptotically polynomial dependence on the process position. Almost all considered cases have no the large deviation principle, but there is a special case, where the principle is established: the rate of a downward jump (death) is growing asymptotically linearly with the population size, while the rate of an upward jump (birth) is growing sub-linearly. We also consider various forms of scaling the original process and normalizing the logarithm of the Large deviation probabilities. The results show interesting features of dependence of the Large deviation functional on the parameters of the underlying process.

Weakly constrained-degree percolation on the hypercubic lattice

Bernardo de Lima (UFMG)

Resumo: We consider the constrained-degree percolation model on the hypercubic lattice, $\mathbb L^d=(\mathbb Z^d,\mathbb E^d)$ for d>=3. It is a continuous time percolation model defined by a sequence, $(U_e)_{e\in\mathbb E^d}$, of i.i.d. uniform random variables in [0,1] and a positive integer (constraint) k. Each bond $e\in\mathbb E^d$ tries to open at time $U_e$; it succeeds if and only if both its end-vertices belong to at most k-1 open bonds at that time.

Our main results are quantitative upper bounds on the critical time, characterising a phase transition for all d>=3 and most nontrivial values of k. As a byproduct, we obtain that for large constraints and dimensions the critical time is asymptotically 1/(2d). For most cases considered it was previously not even established that the phase transition is nontrivial. One of the ingredients of our proof is an improved upper bound for the critical curve, $s_{\mathrm{c}}(b)$, of the Bernoulli mixed site-bond percolation in two dimensions, which may be of independent interest. Joint work with Ivailo Hartarsky.

Quantitative particle approximation of nonlinear Fokker-Planck equations with singular kernel

Christian Olivera (UNICAMP)

Resumo: In this work, we study the convergence of the empirical measure of interacting particle systems with singular interaction kernels. First, we prove quantitative convergence of the time marginals of the empirical measure of particle positions towards the solution of the limiting nonlinear Fokker-Planck equation. Second, we prove the well-posedness for the McKean-Vlasov SDE involving such singular kernels and the convergence of the empirical measure towards it (propagation of chaos).

Our results only require very weak regularity on the interaction kernel, which permits to treat models for which the mean field particle system is not known to be well-defined. For instance, this includes attractive kernels such as Riesz and Keller-Segel kernels in arbitrary dimension. For some of these important examples, this is the first time that a quantitative approximation of the PDE is obtained by means of a stochastic particle system. In particular, this convergence still holds (locally in time) for PDEs exhibiting a blow-up in finite time. The proofs are based on a semigroup approach combined with a fine analysis of the regularity of infinite-dimensional stochastic convolution integrals.

This work is in collaboration with A. Richard and M. Tomasevic.

Results on the contact process with dynamic edges or under renewals

Daniel Ungaretti (UFRJ)

Resumo: I will talk about two variants of the contact process that are built by modifying the percolative structure given by the graphical construction: the Contact Process on Dynamic Edges intro- duced by Linker and Remenik and a generalization of the Renewal Contact Process introduced by Fontes, Marchetti, Mountford and Vares. In recent joint works with Fontes, Mountford and Vares, and with Hilário, Valesin and Vares there has been progress in the understanding of the phase transition of such models. In particular, we will discuss a multiscale renormalization argument that is behind our results regarding almost sure extinction and can possibly be extended to contact processes in some dynamic.random environments.

The contact process over a dynamical d-regular graph

Daniel Valesin (University of Warwick)

Resumo: We consider the contact process on a dynamic graph defined as a random d-regular graph with a stationary edge-switching dynamics. In this graph dynamics, independently of the contact process state, each pair {e1,e2} of edges of the graph is replaced by new edges {e′1,e′2} in a crossing fashion: each of e′1,e′2 contains one vertex of e1 and one vertex of e2. As the number of vertices of the graph is taken to infinity, we scale the rate of switching in a way that any fixed edge is involved in a switching with a rate that approaches a limiting value v, so that locally the switching is seen in the same time scale as that of the contact process. We prove that if the infection rate of the contact process is above a threshold value λ_c (depending on d and v), then the infection survives for a time that grows exponentially with the size of the graph. By proving that λ_c is strictly smaller than the lower critical infection rate of the contact process on the infinite d-regular tree, we show that there are values of λ for which the infection dies out in logarithmic time in the static graph but survives exponentially long in the dynamic graph. Joint work with Gabriel Leite Baptista da Silva and Roberto I. Oliveira.

Detecção de comunidades no modelo estocástico de blocos por máxima verossimilhança

Florencia Leonardi (USP)

Resumo: Nesta palestra pretendo apresentar os resultados fundamentais na literatura sobre estimação das comunidades no modelo estocástico de blocos (SBM, do inglês Stochastick Block Model) com k comunidades, não necessariamente simétrico (todas as comunidades do mesmo tamanho), focando principalmente no método de máxima verossimilhança. Essa abordagem foi considerada anteriormente por Chen e Bickel (2009), mas a prova da consistência do estimador de máxima verossimilhança apresenta alguns pontos ainda em aberto e não totalmente justificados, como apontado em van der Pas e van der Vaart (2018). Neste trabalho, mostramos, usando diferentes desigualdades de concentração, que o estimador de máxima verossimilhança é consistente acima do limiar de transição de fase, para redes com regime de grau logarítmico, completando a prova de Chen e Bickel (2009) e generalizando esses resultados. Este é um trabalho conjunto com Andressa Cerqueira (UFSCAR).

Universality for a class of statistics of hermitian random matrices and the integro-differential Painlevé II equation

Guilherme Silva (USP-São Carlos)

Resumo: It has been known since the 1990s that fluctuations of eigenvalues of random matrices, when appropriately scaled and in the sense of one-point distribution, converge to the Airy2 point process in the large matrix limit. In turn, the latter can be described by the celebrated Tracy-Widom distribution.

In this talk we discuss recent findings of Ghosal (MIT) and myself, showing that certain statistics of eigenvalues also converge universality to appropriate statistics of the Airy2 point process, interpolating between a hard and soft edge of eigenvalues. Such found statistics connect also to the integro-differential Painlevé II equation, in analogy with the celebrated Tracy-Widom connection between Painlevé II and the Airy2 process.

Existence of solution and localization for the stochastic heat equation with multiplicative Lévy white noise

Hubert Lacoin (IMPA)

Resumo: We consider the following stochastic partial differential equation in $\mathbb R^d$

$$ \partial_t u = \Delta u + \xi \cdot u $$

where the unknown u is a function of space and time. The operator $\Delta$ denotes the usual Laplacian in $\mathbb R^d$ and $\xi$ is a space-time Lévy white noise. This equation has been extensively studied in the case where $\xi$ is a Gaussian White noise. In that case, it is known that the equation is well posed only when the space dimension d is equal to one. In our presentation, we consider the case where $\xi$ is a Lévy white noise with no diffusive part and only positive jumps. We identify necessary and sufficient conditions on the Lévy measure $\lambda$ associated with $\xi$ for having existence and uniqueness of solutions to the equation. In dimension one and two the necessary condition and the sufficient one are the same while for d >= 3 they differ only by a third order factor.

Soliton decomposition of a Brownian path

Inés Armendáriz (UBA-CONICET)

Resumo: The Box Ball System, or BBS for short, was introduced by Takahashi and Satsuma in 1990 as a cellular automaton that exhibits solitons (travelling waves). In a recent work, Ferrari, Nguyen, Rolla and Wang propose a hierarchical decomposition of a fixed configuration of the BBS in solitons, called the slot decomposition, and Ferrari and Gabrielli identified the distribution of this decomposition for a random walk with negative drift. In this project we extend these results to a Brownian motion with negative drift. We consider the excursions over past minima of the trajectory, and show that they can be decomposed as a superposition of solitons. These are distributed as a Poisson process in the first quadrant of the plane, with an intensity that is homogeneous in the abscissa (associated to the location of the solitons) but not in the ordinate (denoting the size of the solitons).

Ongoing work with Pablo Blanc, Pablo Ferrari and Davide Gabrielli

Random walk in a birth-and-death dynamical environment

Luiz Renato Fontes (USP)

Resumo: We consider a particle moving in continuous time as a Markov jump process; its discrete chain is given by an ordinary random walk on Z^d (with a 2nd moment), and its jump rate at (x,t) is given by a fixed function f of the state of a birth-and-death (BD) process at x on time t. BD processes at different sites are independent and identically distributed, and f is assumed non increasing and vanishing at infinity. We present an argument to obtain a CLT for the particle position when the environment is 'sufficiently ergodic' (meaning roughly that the BD's equilibrium distribution has a 2nd moment). In the absence of a viable uniform lower bound for the jump rate, we resort instead to stochastic domination, as well as to a subadditive argument to control the time spent by the particle to give n jumps (both ingredients rely on the monotonicity of f); and we also impose conditions on the initial (product) environmental initial distribution. We also present results on the asymptotic form of the environment seen by the particle.

Joint work with Maicon Pinheiro and Pablo Gomes.

Percolation on randomly stretched lattice

Marcelo Hilário (UFMG)

Resumo: We revisit a model of percolation on a random lattice first studied in [Hoffman CMP 254, 1-22, 2005]. Starting from the usual square lattice, rows and columns are selected uniformly at random. All the edges along a selected row or column are removed at once, giving rise to a randomly stretched version of the square lattice. Given a realization of the lattice, independent site percolation is performed. The goal of the present talk is to present a proof for the phase transition for this model. Our proof shares some similarities with the original proof by Hoffman and with the ideas in [Kesten, Sidoravicius, Vares, EJP 27, 1-49 2022] but also brings some new ideas that we expect to be useful in new situations. Based on a joint ongoing work with Marcos Sá, Augusto Teixeira and Remy Sanchis.

On some stochastic models for information transmission on graphs

Pablo Martín Rodríguez (UFPE)

Resumo: In this talk we will discuss recent results related to special stochastic processes to illustrate the phenomenon of information spreading on graphs. Some of the considered models are modified versions of the well known Maki-Thompson rumor model, which is one of the first mathematical models proposed in the literature to describe, in a simple way, the spread of a rumor through an homogeneously mixed population.

Além das palestras indicadas acima, haverá uma sessão de pôsteres. O resumo dos trabalhos apresentados nesta sessão podem ser vistos no link abaixo:

Resumos da sessão de pôsteres.