Abstract: de Lima, Rolla and Valesin (2019) introduced and studied the 2-edges percolation model on homogeneous oriented trees $\mathbb T_d$. In this model, small (length one) edges are opened with probability $p$ and long (length $k>1$) edges are opened with probability $q$. Here, in the slightly more general setting of m-edges ($m\ge2$), we obtain the critical surface as the set of $0$'s of a specific polynomial with coefficients depending on $d$. Specializing to the original 2-edges case, we can extract bounds for the critical parameters, which are valid for any $d$ and $k$.

Abstract: 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 [FG+19]) and the non-cascading case. The cascading occurs when the ground states energies are achieving by adding up the ground states energies of the two levels. In the non-cascading case the correlations are to weak to have an impact on the extremes and the system ”collapses” to a Random Energy Models (REM), see [BK04]. 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.

Abstract: How far is the joint law of a two-dimensional random vector from the product measure induced by its marginals? In this talk we address this question in the context of a Markov process within the KPZ universality class, where the first coordinate of the vector is given by an observable of a Brownian initial condition, and the second one is an observable of the process at a later time. To attack this task we will use tools from Malliavin calculus and Stein’s Method, which will allow us to get a precise space-time scaling behavior for asymptotic independence.

Abstract: We study noise sensitivity of the consensus opinion of the voter model on finite graphs with respect to noise affecting the initial opinions and noise affecting the dynamics. We prove that the final opinion is stable with respect to small perturbations of the initial configuration and is sensitive to perturbations of the dynamics governing the evolution of the process. This talk is based on a joint work with G. Amir, O. Angel, and R. Peretz.

Abstract: Discrepancy between the results of electoral intentions carried out a few days before the actual voting and the electoral poll results during the first round of the 2018 presidential elections in Brazil was striking. At the time, it was conjectured that this discrepancy was the result of social-media campaigning days before the elections. This conjecture raises a question: is social-media campaigning enough to change in a quite short period of time the voting intention of a significant portion of voters? Providing an answer to this question was the initial motivation to introduce a new stochastic model that mimics some important features of real world social networks.

The model for a highly polarized social network is a system of interacting marked point processes. Each point process indicates the successive times in which a social actor expresses a “favorable” (+1) or “contrary” (-1) opinion on a certain subject. The orientation and the rate at which an actor expresses an opinion is influenced by the social pressure exerted on them, modulated by a polarization coefficient. The polarization coefficient of the network indicates the tendency of social actors to express an opinion in the same direction of the social pressure exerted on them. The social pressure on an actor is reset to 0, when they express an opinion, and simultaneously the social pressures on all the other actors change by one unit in the direction of the opinion that was just expressed.

We prove that when the polarization coefficient diverges, this social network reaches consensus instantaneously. Moreover, this consensus has a metastable behavior. This means that the direction of the social pressures on the actors globally changes after a long and unpredictable random time.

This talk is based on a joint work with Antonio Galves.

Abstract: We study a system of independent random walkers in one dimension that annihilate immediately when two particles meet on the same site. In addition, pairs of particles are created randomly on neighbouring sites. For periodic boundary conditions, a duality with independent two-level systems which arises from the integrability of the model is proved. The duality function is determinantal and has the form of a matrix product. We use this duality to compute the exact current distribution. For reflecting boundaries the Markov generator commutes with the generators of a subalgebra of the universal enveloping algebra of the Lie superalgebra sl(1|1) and its deformations. The supersymmetry leads to a duality between an even and odd number of particles, respectively.

Abstract: In 1982, Fröhlich and Spencer, using ideas introduced by the same authors in the study of BKT transition, solved a conjecture by Dyson showing that the one dimensional long-range Ising model with decay undergoes a phase transition for small temperatures. After that, Cassandro, Ferrari, Merola, and Presutti provided a contour argument to prove phase transition for the same model with more general decays. In this talk, we will show how a contour argument can be developed to show phase transition for multidimensional long-range Ising models. As an application, we will show how our techniques apply to the model with a decaying field.

The talk is based on joint work with Rodrigo Bissacot, Eric O. Endo, and Satoshi Handa.

Abstract: The KPZ Universality says that particle systems with certain spatio-temporal correlation structures display universal fluctuations in the appropriate scaling limits. We aim to discuss some results that appear in the context of KPZ Universality. Our toy models will be the classical totally asymmetric simple exclusion process (TASEP) and its periodic version. We survey how the celebrated Tracy-Widom distribution arises in the classical TASEP and how it is described in terms of a nonlinear ODE. In contrast, the limiting distribution for the periodic TASEP was only recently obtained, and we plan to explain how it relates to certain integrable systems that arise in nonlinear wave theory.

The talk is based on joint work with Jinho Baik and Zhipeng Liu.

Abstract: In this talk, I will discuss some recent progress on the limiting behavior of random walks on dynamic random environments. We will mainly consider stationary one-dimensional random environments that exhibit bad mixing conditions, meaning that the mixing rates are not uniform over the initial configuration.

We will show how renormalization can be used to deal with this class of models leading to results such as law of large numbers and large deviation estimates. Under certain conditions, with the aid of a regeneration structure, we are also able to deduce central limit theorems. The talk is based on several joint works with Oriane Blondel, Frank den Hollander, Daniel Kious, Renato dos Santos, Vladas Sidoravicius.

Abstract: We consider a continuous time symmetric random walk on the integers,

whose rates are equal to 1/2 for all bonds, except for the bond

of vertices {−1, 0}, which associated rate is given by \alpha n^{-\beta}/2 , where \alpha and \beta

are parameters of the model. We prove here a functional central

limit theorem for the random walk with a slow bond: if \beta<1, then it con-

verges to the usual Brownian motion. If \beta>1, then it converges to the

reflected Brownian motion. And at the critical value \beta = 1, it converges to the

snapping out Brownian motion (SNOB) of parameter k = 2 \alpha, which is a Brow-

nian type-process recently constructed by Lejay (2016). We also provide Berry-Esseen

estimates in the dual bounded Lipschitz metric for the weak convergence of

one-dimensional distributions, which we believe to be sharp.

Talk based on a joint work with D. Erhard and D. Silva.

Abstract: We will discuss the contact process, a model for the spread of an infection in a population, on random graph models in which the degree distribution is a power law. In such graphs, the contact process exhibits metastable behavior (that is, the infection stays active for a very long time) even if the infection rate is close to zero. We will focus on two such random graph models: the configuration model and random hyperbolic graphs. In both these cases, we discuss aspects of the behavior of the process, including the distribution of the extinction time of the infection and the density of infected vertices in typical times of activity. We show in particular that the critical exponent of this density, as the infection rate is taken to zero, is the same for both random graph models, suggesting some universality phenomenon. We will touch on joint work with Amitai Linker, Dieter Mitsche, Thomas Mountford, Jean-Christophe Mourrat, Bruno Schapira and Qiang Yao.

Abstract: We consider some problems related to the truncation question in long-range percolation. Probabilities are given 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 under which the answer is affirmative. One of these results generalizes a previous result in [Alves, Hilário, de Lima, Valesin, Journ. Stat. Phys. 122, 972 (2017)]. Joint work with Alberto M. Campos

Abstract: We introduce a one-parameter family of critical Galton-Watson tree measures invariant under the operation of Horton pruning (cutting tree leaves followed by series reduction). Under a regularity condition, this family of measures are the attractors of critical Galton-Watson trees under consecutive Horton pruning. The invariant Galton-Watson (IGW) measures with i.i.d. exponential edge lengths are the only Galton-Watson measures invariant with respect to all admissible types of generalized dynamical pruning (an operation of erasing a tree from leaves down to the root).

This is a joint work with Ilya Zaliapin (University of Nevada Reno) and Guochen Xu (Oregon State University).

Abstract: We study an SIS epidemic model with infections carried by mobile agents performing independent random walks on a graph. Agents can either be infected (I) or susceptible (S), and an infection occurs when an infected agent meets a susceptible one. After a recovery time, an infected agent returns to state S and can be infected again. The End of Epidemic (EoE) denotes the first time when all agents are in state S, since after this moment no further infection can occur. We present some results for the case of two agents on edge-transitive graphs. Specifically, we characterize EoE as a function of the network structure by relating the Laplace transform of EoE to the Laplace transform of the meeting time of two random walks. We also study the asymptotic behavior of EoE (asymptotically in the size of the graph) on complete graphs, complete bipartite graphs, and rings.

This is joint work with Seva Shneer (Heriot-Watt University) and Daniel Figueiredo (COPPE/UFRJ).

Abstract: A well-known phenomenon for independent percolation on the d-dimensional hipercubic lattice is that, in some sense, for d large, it resembles the model on the (d+1)-regular tree.

In this talk we will discuss this phenomenon for anisotropic percolation models. On the oriented case, we will show that, if the sum of the local probabilities is strictly greater than one and each of them is not too large, then percolation occurs. We also will show that, in high dimensions, the crossover critical exponent is equal to one, the same value known for regular trees.

Based on joint works with Alan Pereira, Rémy Sanchis and Roger W.C. Silva.

Abstract: In this seminar we present some limit theorems according to Wasserstein distance, both for positive associated: stationary and non-stationary random processes. It is well known that the convergence in Wasserstein distance is stronger than that of distribution, therefore, implicitly we will also study Central Limit theorems. As an application of our results, we will display some one-dimensional random processes with Gibbsian dependence that converge in Wasserstein distance.

Abstract: In this talk I will introduce a multidimensional walk with memory and random tendency. I will prove a law of large numbers and the existence of a phase transition from diffusive to superdiffusive regimes. In the first case, it is possible to obtain a functional limit theorem to Gaussian vectors. In superdiffusive regime, we obtain strong convergence to a non-Gaussian random vector and characterize its moments.

Abstract: We propose a mathematical model to measure how multiple repetitions may influence the ultimate proportion of the population never hearing a rumor during a given outbreak. The model is a multi-dimensional continuous-time Markov chain that can be seen as a generalization of the Maki--Thompson model for the propagation of a rumor within a homogeneously mixing population. In the well-known basic model, the population is made up of ``spreaders'', ``ignorants'' and ``stiflers'', and any spreader attempts to transmit the rumor to the other individuals via directed contacts. In case the contacted individual is an ignorant, it becomes a spreader, while in the other two cases the initiating spreader turns into a stifler. The process in a finite population will eventually reach an equilibrium situation, where individuals are either stiflers or ignorants. We generalize the model by assuming that each ignorant becomes a spreader only after hearing the rumor a predetermined number of times. We identify and analyze a suitable limiting dynamical system of the model, and we prove limit theorems that characterize the ultimate proportion of individuals in the different classes of the population.

This is a join work with Cristian Coletti, Elcio Lebensztayn and Pablo M. Rodriguez.

Abstract: 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.

Abstract:The KPZ equation is the stochastic partial differential equation in d space dimensions formally given by \partial_t h=\Delta h +\langle h,Q h\rangle +\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

Abstract: We consider the positive solution to the heat equation with random multiplicative potential on the d-dimensional lattice. We will discuss geometric aspects of the intermittency phenomenon, in which the total mass of the solution concentrates asymptotically for large times in ''islands'' that are relatively small and well-separated in space. We will also discuss extensions of the model and interpretations in terms of population genetics.

Abstract: 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.

Abstract: It was recently proved that the set of DLR-Gibbs measures, associated with a uniformly absolutely summable interaction on the lattice N, coincides with the set of P(f)-conformal measures associated with a suitable continuous potential f. In this talk, we explore this result and prove a new relation between the set of extreme P(f)-conformal measures and the dimension of the Perron-Frobenius eigenspace of the L1-extension of the transfer operator associated with the potential f. In particular, we show that such eigenspace's geometric multiplicity can only be greater than one when it occurs a first-order phase transition in Dobrushin's sense.

We obtain this result by looking at the transfer operator's extension as a Markov Process in the Hopf's sense. The harmonic analysis we will present is based on the convexity properties of the set of P(f)-conformal measures. We also show that if the Perron-Frobenius eigenspace dimension is one and the potential satisfies additional regularity properties, then a Functional Central Limit Theorem holds for equilibrium states and non-local observables.

Abstract: We will discuss the two-dimensional simple random walk conditioned on never hitting the origin, which is, formally speaking, the Doob's $h$-transform of the simple random walk with respect to the potential kernel. This random walk is the main building-block of the construction of random interlacements on the plane introduced by Comets, Popov and Vachkovskaia. However, this walk has become an interesting object on its own. To justify this claim we present a few of its properties, citing some of the current literature and presenting the results of a recent joint work with Serguei Popov (Unicamp/CMUP) and Leonardo Rolla (UBA/NYU Shangai).