Seminários de Probabilidade 2021 - 2º Semestre


We introduce the Drainage Network with Branching, which is a system of coalescing random walks with paths that can branch and that exhibit some dependence before coalescence. It extends the Drainage Network model introduced by Gangopadhyay, Roy and Sarkar in 2004, by allowing the paths to branch. We also study the convergence of the Drainage Network with Branching, under diffusive scaling, to the Brownian Web or Net, according to specific conditions for the branching probability. We show that based on the specification of the branching probability, we can have convergence to the Brownian Web or we can have a tight family such that any weak limit point contains a Brownian Net. In the latter case, we conjecture that the limit is indeed the Brownian Net. This is a joint work with Glauco Valle (IM-UFRJ) and Leonel Zuaznabar (IME-USP).


Discrete Markov random fields on graphs, also known as graphical models in the statistical literature, have become popular in recent years due to their flexibility to capture conditional dependency relationships between variables. They have already been applied to many different problems in different fields such as Biology, Social Science, or Neuroscience. Graphical models are, in a sense, finite versions of general random fields or Gibbs distributions, classical models in stochastic processes. This talk will present the problem of estimating the interaction structure (conditional dependencies) between variables by a penalized pseudo-likelihood criterion. First, I will consider this criterion to estimate the interaction neighborhood of a single node, which will later be combined with the other estimated neighborhoods to obtain an estimator of the underlying graph. I will show some recent consistency results for the estimated neighborhood of a node and any finite sub-graph when the number of candidate nodes grows with the sample size. These results do not assume the usual positivity condition for the conditional probabilities of the model as it is usually assumed in the literature of Markov random fields. These results open new possibilities of extending these models to situations with sparsity, where many parameters of the model are null. I will also present some ongoing extensions of these results to processes satisfying mixing type conditions. This talk is based on a joint work with Iara Frondana and Rodrigo Carvalho and some work in progress with Magno Severino.


We look at the contact process with ordinary rate lambda exponential infections and heavy tailed cures, attracted to an alpha-stable law with alpha < 1, on a finite graph of size k. Our aim is to ascertain conditions on alpha and k such that the critical lambda for survival of the infection vanishes. We obtain nearly sharp (in a sense to be clarified) bounds on the critical k, k_c = k_c(alpha), which is always a finite number, such that the infection dies out almost surely for any lambda < infty at and below k_c; and there is positive probability of survival for any lambda > 0 above k_c. This is joint work with Pablo Almeida Gomes and Rémy Sanchis, published recently, in Bernoulli 27(3).

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I present results recently obtained with Francesco Manzo e Matteo Quattropani.

We present an alternative proof of the so-called First Visit Time Lemma (FVTL), originally presented by Cooper and Frieze. We work in the original setting, considering a growing sequence of irreducible Markov chains on n states. We assume that the chain is rapidly mixing and with a stationary measure with no entry being either too small nor too large. Under these assumptions, the FVTL shows the exponential decay of the distribution of the hitting time of a given state x, for the chain started at stationarity, up to a small multiplicative correction. While the proof by Cooper and Frieze is based on tools from complex analysis, and it requires an additional assumption on a generating function, we present a completely probabilistic proof, relying on the theory of quasi-stationary distributions and on strong-stationary times arguments. In addition, under the same set of assumptions, we provide some quantitative control on the Doob’s transform of the chain on the complement of the state x.

I will also discuss the relation of this result with general results, previously obtained, providing an exact formula for the first hitting distribution via conditional strong quasi-stationary times.

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We construct an exclusion process with Bernoulli product invariant measure and having, in the diffusive hydrodynamic scaling, a non symmetric diffusion matrix, that can be explicitly computed. The antisymmetric part does not affect the evolution of the density but it is relevant for the evolution of the current. In particular because of that, the Fick’s law is violated in the diffusive limit. Switching on a weak external field we obtain a symmetric mobility matrix that is related just to the symmetric part of the diffusion matrix by the Einstein relation. We show that this fact is typical within a class of generalized gradient models. We consider for simplicity the model in dimension $d=2$, but a similar behavior can be also obtained in higher dimensions.

Joint work with L. De Carlo and P. Goncalves.

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In this talk we will briefly present the model we are interested in, which is a fractional elliptic stochastic partial differential equation driven by Gaussian white noise. There is in the literature a standard way to approximate the covariance operator of the solution of such equations, the so-called rational approximation (Bolin and Kirchner, 2020), however this approach uses the solution to build such an approximation. By considering directly the covariance operator, we are able to provide a more computationally efficient approximation. We compute the rate of this approximation in terms of the Hilbert-Schmidt norm. Furthermore, we also obtain, rigorously, the rate of approximation of the so-called lumped mass method. This method is widely used by practitioners and is essential to make it computationally feasible to fit some models in spatial statistics. We obtain the rate of approximation of the lumped mass method in terms of the operator’s norm as well as, under some additional restrictions, the Hilbert-Schmidt norm. Finally, we present the usage of these approximations in maximum likelihood estimation. Joint work with David Bolin and Zhen Xiong.

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I discuss the low-temperature behaviour of Dyson models (polynomially decaying long-range Ising models in one dimension) in the presence of random boundary conditions. As for typical random (i.i.d.) boundary conditions Chaotic Size Dependence occurs, that is, the pointwise thermodynamic limit of the finite-volume Gibbs states for increasing volumes does not exist, but the sequence of states moves between various possible limit points, as a consequence it makes sense to study distributional limits, the so-called “metastates” which are measures on the possible limiting Gibbs measures.

The Dyson model is known to have a phase transition for decay parameters α between 1 and 2. We show that the metastate changes character at α =3/2. It is dispersed in both cases, but it changes between being supported on two pure Gibbs measures when α is less than 3/2 to being supported on mixtures thereof when α is larger than 3/2.

Joint work with Eric Endo and Arnaud Le Ny.

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We introduce the equilibrium Widom-Rowlinson model on a two-dimensional finite torus in which the energy of a particle configuration is attractive and determined by the union of small discs centered at the positions of the particles. We then discuss the metastable behaviour of a dynamic version of the WR model. This means that the particle configuration is viewed as a continuous time Markov process where particles are randomly created and annihilated as if the outside of the torus were an infinite reservoir with a given chemical potential. In particular, we start with the empty torus and are interested in the first time when the torus is fully covered by discs in the regime at low temperature and when the chemical potential is supercritical. In order to achieve the transition from empty to full, the system needs to create a sufficiently large droplet, called critical droplet, which triggers the crossover. We compute the distribution of the crossover time and identify the size and the shape of the critical droplet. The analysis relies on a mesoscopic and microscopic description of the surface of the critical droplet. It turns out that the critical droplet is close to a disc of a certain deterministic radius, with a boundary that is random and consists of a large number of small discs that stick out by a small distance. We will show how the analysis of surface fluctuations in the WR model allows us to derive the leading order term of the condensation time and also the correction order term. This is a joint work with Frank den Hollander (Leiden), Sabine Jansen (Munich) and Roman Kotecky (Prague & Warwick).

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23/08 (excepcionalmente às 14h)

The Parabolic Anderson Model on a Galton-Watson Tree – Frank den Hollander (Leiden University)

We consider the parabolic Anderson model on a supercritical Galton-Watson tree with an i.i.d. random potential whose marginal distribution is close to the double exponential. Under the assumption that the degree distribution has a sufficiently thin tail, we derive an asymptotic expansion for large times of the total mass of the solution given that initially a unit mass sits at the root. We derive the expansion both under the quenched law (i.e., conditional on the realisation of the random tree and the random potential) and under the half-annealed law (i.e., conditional on the realisation of the random tree but averaged over the random potential). The two expansions turn out to be different, but both contain a coefficient that is given by a variational formula indicating that the solution concentrates on a subtree with minimal degree according to a computable profile. A key tool in the analysis is the large deviation principle for the empirical distribution of a Markov renewal process. Joint work with Wolfgang König (Berlin), Renato dos Santos (Belo Horizonte), Daoyi Wang (Leiden).

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Local Scaling Limits of Lévy Driven Fractional Random Fields – Donatas Surgailis (Vilnius University)

We obtain a complete description of local anisotropic scaling limits for a class of fractional random fields $X$ on ${mathbb{R}}^2$ written as stochastic integral with respect to an infinitely divisible random measure. The scaling procedure involves increments of $X$ over points the distance between which in the horizontal and vertical directions shrinks as $O(lambda) $ and $O(lambda^gamma)$ respectively as $lambda downarrow 0$, for some $gamma>0$. We consider two types of increments of $X$: usual increment and rectangular increment, leading to the respective concepts of $gamma$-tangent and $gamma$-rectangent random fields. We prove that for above $X$ both types of local scaling limits exist for any $gamma>0$ and undergo a transition, being independent of $gamma>gamma_0$ and $gamma<gamma_0$, for some $gamma_0>0$; moreover, the `unbalanced' scaling limits ($gamm negamma_0$) are $(H_1,H_2)$-multi self-similar with one of $H_i$, $i=1,2$, equal to $0$ or $1$. The paper extends Pilipauskait.e and Surgailis (2017) and Surgailis (2020) on large-scale anisotropic scaling of random fields on ${mathbb{Z}}^2$ and Benassi et al. (2004) on $1$- tangent limits of isotropic fractional Lévy random fields. This is joint work with Vytautė Pilipauskaitė (University of Luxembourg).

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Real-world networks are often understood as being symmetrical, meaning that vertices can be found which perform similar or equivalent structural roles (such as hubs from different communities in social networks, or functional regions in neuronal networks). These roles are usually associated with their topological placement relative to its surroundings; however, traditional mathematical formulations of graph symmetry are based on automorphism groups, which depend fundamentally on global structure and do not account for similarities in local structures. In this work, we introduce the concept of local symmetry, which reflects the structural equivalence of vertices’ egonets while generalizing classical conceptualizations of symmetry such as automorphism and isomorphism. We also study the emergence of local asymmetry in Erdős–Rényi graphs, identifying regimes of both asymptotic local symmetry and asymptotic local asymmetry. We find that local symmetry persists at least to an average degree of n^{1/3} and local asymmetry emerges at an average degree not greater than n^{1/2}, which are regimes of much larger average degree than for traditional, global asymmetry.

Joint work with Daniel Figueiredo (COPPE/UFRJ) and Valmir Barbosa (COPPE/UFRJ).


We consider a generalised oriented site percolation model on Z^d with arbitrary neighbourhood. The key additional difficulties as compared to standard oriented percolation are the lack of symmetry and, in two dimensions, of planarity. We establish that, despite these deficiencies, in the supercritical regime GOSP behaves qualitatively like OP.

Joint work with Ivailo Hartarsky.

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