Posters

Remark: Please bring your poster printed in A0 format.



Tuesday 10h15

  1. Alan Anderson da Silva Pereira (UFAL),
    Marked random graphs with given degree sequence: large deviations on the local topology We investigate the behavior of the empirical neighbourhood distribution of marked graphs in the framework of local weak convergence. We establish a large deviation prin- ciple for such families of empirical measures. The proof builds on Bordenave and Caputo’s seminal 2015 paper, and Delgosha and Anantharam’s 2019 introduction of BC entropy, relying on combinatorial lemmas that allow one to construct suitable approximations of measures supported on marked trees.
  2. Alberto González Sanz (Columbia University),
    Weak limits for empirical entropic optimal transport We establish weak limits for the empirical entropy regularized optimal transport cost, the expectation of the empirical plan and the conditional expectation. Our results require only uniform boundedness of the cost function and no smoothness properties, thus emphasizing the far-reaching regularizing nature of entropy penalization. To derive these results, we employ a novel technique that sidesteps the intricacies linked to empirical process theory and the control of suprema of function classes determined by the cost. Instead, we perform a careful linearization analysis for entropic optimal transport with respect to an empirical L2-norm, which enables a streamlined analysis. As a consequence, our work gives rise to new implications for a multitude of transport-based applications under general costs, including pointwise distributional limits for the empirical entropic optimal transport map estimator, kernel methods as well as regularized colocalization curves. Overall, our research lays the foundation for an expanded framework of statistical inference with empirical entropic optimal transport.
  3. Alexandre Batista de Souza (UNICAMP),
    Convergence problem with multiplicative noise We solved the convergence problem in mean field games with multiplicative common noise. Additionally, we have a convergence rate for the additive case. The proofs relies in energy methods, since that the semigroup approach doesn't work.
  4. Anatoli Iambartsev (IME-USP),
    Limits for Birth-and-Death Processes with Polynomial Rates The symmetric birth and death stochastic process on the non-negative integers with polynomial rates is studied. The process moves slowly and spends more time in the neighborhood of the state \(0\). We prove the convergence of the scaled process to a solution of stochastic differential equation without drift. Sticking phenomenon appears at the limiting process: trajectories, starting from any state, take finite time to reach 0 and remain there indefinitely.
  5. Bastián Arturo Mora García (PUC-Chile),
    Towards a new proof of the Intermediate Disorder Regime for Directed Polymers In dimension \(1+1\), if the random polymer paths are re-scaled diffusively and, simultaneously, the inverse temperature is re-scaled with the polymer length \(n\) as \(\beta_n := \frac{n^{-1/4}}{\sqrt{2}} \beta\), a particular behavior is observed (called the Intermediate Disorder Regime). A decade ago, Alberts, Khanin and Quastel proved that, under the above scaling, the normalized point-to-point partition function \(Z_{n, \sqrt{n} x}(\omega, \beta_n) e^{n \lambda(\beta_n)} \) of the random polymer converges in distribution to the solution \(\mathcal{Z}_{\beta}(1,x)\) of the Stochastic Heat Equation with multiplicative White Noise \(\xi\), which is classically expressed by its Chaos Expansion. We give a new (partial) proof of this convergence, based on recent results by Quastel, Ramírez and Virág, where they express \(\mathcal{Z}_{\beta}(t,x)\) as the limit of a uniformly integrable martingale of a (natural) filtration of \(\xi\). We prove that certain approximations of \(Z_{n, \sqrt{n} x}(\omega, \beta_n) e^{n \lambda(\beta_n)}\) converge in distribution to the corresponding approximations of \(\mathcal{Z}_1(1,x)\) given by these martingale terms.
  6. Beatriz da Costa Salvador (IST-Lisboa),
    On correlations for partial exclusion, inclusion and independent particles In this talk, we introduce the concept of Stochastic Duality. From it, we will see an approach, based on duality, on how to bound the k-points centered correlation functions of a given interacting particle system (IPS) when it has the duality property and also a specific type of duality function. The problem of finding such bounds is an essential tool to derive the non-equilibrium fluctuations of boundary-driven IPS. We will see an application of those results for the symmetric simple partial exclusion process, SEP(\(\alpha\)), the symmetric simple inclusion process, SIP(\(\alpha\)), and independent particles, IRW, all considered with open boundary. The case \(k = 2\) for SEP(\(\alpha\)) is joint work with Chiara Franceschini, Patrícia Gonçalves and Milton Jara [1] and the general case is a work in progress with Patrícia Gonçalves. [1] Franceschini, C., Gonçalves, P., Jara, M., Salvador, B. (2023): Non-equilibrium fluctuations for SEP(\(\alpha\)) with open boundary, submitted and online at https://arxiv.org/abs/2308.09144.
  7. Berend van Tol (TU Delft),
    Intertwining and propagation of stochastic mixture measures We study a class of stochastic models of mass transport on discrete vertex set \(V\). For these models, a one-parameter family of homogeneous product measures \(\otimes_{i\in V} \nu_\theta\) is reversible. The set of mixtures of inhomogeneous product measures with equilibrium marginals, i.e., the set of measures of the form \[ \int\Big(\bigotimes_{i\in V} \nu_{\theta_i}\Big) \,\Xi(\prod_{i\in V}d\theta_i) \] is left invariant by the dynamics in the course of time, and the ``mixing measure'' \(\Xi\) evolves according to a Markov process which we then call ``the hidden parameter model''. The class of models includes discrete and continuous generalized KMP models, as well as discrete and continuous harmonic models. The results imply that in all these models, the non-equilibrium steady state of their reservoir driven version is a mixture of product measures where the mixing measure is in turn the stationary state of the corresponding ``hidden parameter model''. For the boundary-driven harmonic models on the chain \(\{1,\ldots, N\}\) with nearest neighbor edges, we recover that the stationary measure of the hidden parameter model is the joint distribution of the ordered Dirichlet distribution, with a purely probabilistic proof based on a spatial Markov property of the hidden parameter model.
  8. Carlos David Martinez Arevalo (IMCA),
    Discrete Poisson traps and the detection model Let the particles from a homogeneous Poisson Point Process (PPP) in the lattice \(\mathbb{Z}^d\) move as discrete time simple random walks. There is a target at \(0\) that doesn't want to be detected by these particles and is clairvoyant, this is that it can see the future. We prove that there is a non-trivial phase transition that depends on the intensity of the PPP for \(d > 1\). The proof relies on multi-scale analysis and a directed random interlacements representation. We also discuss some open problems related to the subcritical phase and the existence of scape when for each time there are only finite open clusters.
  9. Célio Augusto Terra de Souza (UFMG),
    On the slow phase for fixed energy ARW We study Activated Random Walks on the one-dimensional ring. We prove that, for every sleeping rate, with high probability the system will take a exponentially large time to stabilize if the density of particles is close enough to \(1\), using estimates about the tail of simple random walks.
  10. Chan Yoon Jun, (Universität Augsburg),
    Correlation decay for particle systems on the real line interacting via a Lennard-Jones type potential Decay of correlation plays an important role in statistical physics. While huge number of results are obtained for lattice systems, results for particle systems in the continuum are lacking. We study the decay of correlation for particle systems on the real line using the theory of random point process.

Tuesday 16h

  1. Cristian Coletti (UFABC),
    Two repelling random walks on \(\mathbb Z\) We consider two interacting random walks on \(\mathbb Z\) such that the transition probability of one walk in one direction decreases exponentially with the number of transitions of the other walk in that direction. The joint process may thus be seen as two random walks reinforced to repel each other. The strength of the repulsion is further modulated in our model by a parameter \(\beta\geq 0\). When \(\beta =0\) both processes are independent symmetric random walks on \(\mathbb Z\), and hence recurrent. We show that both random walks are further recurrent if \(\beta \in (0,1]\). We also show that these processes are transient and diverge in opposite directions if \(\beta >2\). Joint work with Fernando Prado (USP) and Rafael Rosales (USP).
  2. Christian Jaime Maura Llauri (IMPA),
    \(\Gamma \) - Expansion of non-reversible difusions over the one-dimensional torus Consider the one-dimensional diffusion process with periodic boundary conditions \[ dX_{\epsilon}(t) = b(X_{\epsilon}(t))dt + \sqrt{2\epsilon}dW_{t}\] where \(W_{t}\) is a Brownian motion on the torus \(\mathbb{T} = [0,1)\) and \(b\) is a drift of class \(C^{2}(\mathbb{T})\) with a finite number critical points, each of them being non-degenerate. Assume that \(\int_{0}^{1}b(\theta)d\theta > 0\), so that the process \(X_{\epsilon}(t)\) is non-reversible. Denote by \(P(\mathbb T)\) the set of probability measures on \(\mathbb T\), endowed with the weak topology, and by \(J_{\epsilon}: P(\mathbb T)\to [0,+\infty)\) the level two large deviation rate functional for \(X_{\epsilon}(t)\) as \(t\to\infty\). We derive a full \(\Gamma-\)expansion of \( J_{\epsilon}\), expressing it as \(J_\epsilon = \epsilon \widehat{I} + I^{(0)} + \sum_{p=1}^{\mathfrak q}\frac{1}{\theta^{p}_{\epsilon}}\; I^{p}\), where \(\widehat{ J},\; J^p: P(\mathbb{T})\to [0,+\infty]\) represent rate functionals independent of \(\epsilon\) and \(\theta^{p+1}_{\epsilon}\) are sequences satisfying \(\theta^{p+1}_{\epsilon} \to +\infty\) and \(\theta^{p+1}/\theta^{p+2}\to 0\) for \(0\leq p<\mathfrak{q}-1\). Each sequence \(\theta^{p}_{\epsilon}\) corresponds to the time-scale at which the Markov process \(\mathbb{X}_{\epsilon}(t)\) exhibits metastable behaviour.
  3. Claudia Lorena Duarte Espitia (UNICAMP),
    Stochastic nonlinear partial differential equations in the space of almost periodic functions We will talk about the well-posedness and the long-time behavior of almost periodic solutions to stochastic degenerate parabolic-hyperbolic equations in any space dimension, under the assumption of Lipschitz continuity of the flux and viscosity functions and a non-degeneracy condition. In particualr, we will show the existence and uniqueness of an invariant measure in a separable subspace of the space of Besicovitch almost periodic functions. This is a joint work with Hermano Frid and Daniel Marroquin.
  4. Daniel Miranda Machado (UFABC),
    Teoremas limites para a homologia persistente TBA.
  5. Daniel Ungaretti Borges (UFRJ),
    Oriented percolation with inhomogeneities and strict inequalities We discuss natural questions for oriented percolation on a layered environment that introduces long range dependence. Layers are independently considered to be bad or good, with bad layers being harder to cross. We focus on a graph known as the hexagonal space lattice, which can be seen as a \(3\)-dimensional version of \(\mathbb{Z}^2\) with oriented edges. As a convenient tool, we are led to deal with questions on the strict decrease of the percolation parameter in the oriented setup when an extra dimension is added. Joint work with Maria Eulalia Vares (UFRJ) and Bernardo de Lima (UFMG).
  6. Davi Matheus Costa Barros (UFAL),
    Prevendo a trajetória de um processo aparentemente imprevisível O objetivo deste trabalho é introduzir uma abordagem simples e construtiva do movimento browniano, utilizando exemplos discretos e contínuos para esclarecer o teorema de extensão de Kolmogorov. Através de tópicos iniciais de probabilidade e processos estocásticos, estuda-se o comportamento das partículas em movimento browniano, proporcionando uma análise desse fenômeno aparentemente imprevisível.
  7. Denis Araujo Luiz (UFABC),
    Non Markovian Rumor Models We propose a non-Markovian model of rumor spreading in continuous time on the complete graph of order \(n\) and investigate asymptotic properties as \(n\to\infty\). In the proposed model, there are four classes of individuals: inactive (those who have not been in contact with the rumor), spreaders (those who are spreading the rumor), contestants (those who claim that the rumor should not be spread), and passive (those who have not yet decided whether to spread or contest). The dynamics is as follows: inactive individuals become passive after contacting spreader individuals; after a random time, a passive individual becomes either a spreader or a contestant with a fixed probability; when a spreader individual contacts a contestant, the spreader becomes a contestant; and spreader individuals and contestant individuals become inactive again after random time depending on the class of the individual. This study based on the mean-field analysis of counting processes.
  8. Diogo Carlos dos Santos (UFAL),
    On the number of infinite clusters in the constrained-degree percolation model We consider constrained-degree percolation model on the hypercubic lattice \(L^d=(\mathbb Z^d,\mathbb E^d)\). In this model, there exists a sequence \((U_e: e \in \mathbb E^d)\) of independent and uniformly distributed random variables in the interval \([0,1]\) and a positive integer \(k\), which is called a constraint. Each edge e attempts to open at time \(U_e\), and the attempt is successful if the number of neighboring edges open at each endvertex of e is at most \(k-1\). In 2022, Hartarsky and De Lima demonstrated that this model undergoes a phase transition when \(d\) is greater than 2 and for most nontrivial values of \(k\). In the present work, we prove that, for any fixed constraint, the number of infinite clusters at any given time \(t \in [0,1)\) is either \(0\) or \(1\), almost surely. This is a joint work with Weberson S. Arcanjo and Alan S. Pereira.
  9. Eldon Barros do Reis Junior (IMPA),
    Discrete Entropic Central Limit Theorem The study of theoretical methods of entropy in relation to the Central Limit Theorem dates back to 1959, when Linnik proposed this investigation in its seminal article. Although he demonstrated weak convergence, Linnik left room for a deeper understanding of this phenomenon. In 1986, Barron made a significant contribution to this area by presenting a complete proof of the Central Limit Theorem for random variables with densities, incorporating concepts from Information Theory into his work, as documented in its paper. The relation between the entropy and Central Limit Theorem is intriguing, given that the Gaussian is the one that maximizes entropy under bounded variance. As a result, we may reformulate the Central Limit Theorem, stating that the entropy of convolution of i.i.d real random variables converges to its maximum value. This phenomenon is known as Convergence in Relative Entropy or Entropic Central Limit Theorem. More recently, in 2021, Gavalakis and Kontoyiannis expanded this understanding by demonstrating a discrete analogue of the Entropic Central Limit Theorem for Lattice Random Variables.
  10. Estêvão Ferraz Borel (UFMG),
    Anisotropic Ising model in \(d+s\) dimensions In this work, we consider the asymmetric Ising model on the \((d+s)\)-dimensional cubic lattice, with \(\beta=1\) and coupling constants \(J_s\) and \(J_d\) for edges of \(\mathbb{Z}^s\) and \(\mathbb{Z}^d\), respectively. We obtain a lower bound for the critical curve in the phase diagram of \((J_s,J_d)\). In particular, as \(J_d\) approaches its critical value from below, our result is directly related to the so-called dimensional crossover phenomenon. This is a joint work with A. Procacci, R. Sanchis and R. Silva.

Wednesday 10h30

  1. Facundo Zanola (Leiden University),
    Voter model on sparse random digraphs We consider Markovian dynamics on a typical realization of the so-called Directed Configuration Model (DCM), which is a random directed graph with prescribed in- and out-degrees. In this random geometry, we study the meeting time of two random walks on a typical realization of the graph starting at stationarity, the coalescence time for a system of coalescent random walks, and the consensus time of the voter model. Indeed, it is known that the latter three quantities are related to each other when the underlying sequence of graphs satisfies certain mean field conditions. We provide a complete characterization of the distribution of meeting, coalescence and consensus time on a typical random graph as a function of a single quantity \(\theta\). More precisely we show that, for a typical large graph from the DCM ensemble, the distribution of the meeting time is well- approximated by an exponential random variable. Furthermore, we provide the precise first-order approximation of its expectation, showing that the latter is linear in the size of the graph, and the explicit preconstant \(\theta\) depends on some easy statistics of the degree sequence. As a consequence, we can analyze the effect of the degree sequence on the ex- pected meeting time and, via some explicit examples, how its regularity/variability play crucial roles in the information diffusion. This is based on a joint work with Luca Avena (University of Florence), Rajat Subhra Hazra (Leiden University) and Matteo Quattropani (Sapienza, University of Rome).
  2. Fernando Pigeard de Almeida Prado (DCM-USP),
    Interacting vertex reinforced random walks on complete sub-graphs This article introduces a model for \(m\) interacting vertex-reinforced random walks on a non-complete graph composed of \(m\) complete sub-graphs \(G^i\), where \(i = 1,2,\ldots, m\). Each walk \(i\) transits between vertices \(v\) and \(w\) within its respective complete sub-graph \(G^i\), although the sub-graphs may share common vertices. The transition probability of a walk \(i\) from vertex \(v\) to \(w\) is independent of \(v\) and depends instead on \(w\) and the proportions of past visits to \(w\) by all other walks \(j\), including its own, with these visits weighted differently based on \(w\) and the specific vertices \(j\) that visited it. The model resembles a quite general Polya urn model. We demonstrate that the empirical vertex occupation measure converges almost surely and establish that such convergence is, in fact, a generic behavior. Our findings are illustrated through a few rich examples of competing walks on non-complete star graphs and circles.
  3. Francisco Alan Lima da Silva (UFAL),
    Limiares para a existência de cliques e conexidade no grafo de Erdös-Rényi Os grafos aleatórios foram introduzidos por Erdös e Rényi no século passado. Pouco tempo depois, ainda no mesmo ano, Gilbert introduziu o modelo que aqui será discutido. O objetivo deste trabalho é expor um limiar para a propriedade de existir \(r\)-cliques no grafo de Gilbert-Erdös-Rényi. Também será exposto um limiar súbito para a conexidade do grafo aleatório.
  4. Gabriel dos Reis Trindade (IME-USP),
    Information geometry: when statistics meets geometry TBA.
  5. Gabriel Silva Nahum (Inria-Lyon),
    On the Construction of Gradient Models In the context of Interacting Particle Systems, a model is said to satisfy the "gradient property" if the microscopic current of a conserved quantity can be expressed as the gradient of some local function. This property is often the starting point for studying a model, as it helps to avoid many technical difficulties regarding the scaling limit of the conserved quantity. An unclear and fundamental step, is whether there is a systematic approach to defining a gradient model given certain desired properties of diffusion. In this talk, I will present how we generalized the Porous Media Model, which is a gradient model associated with a diffusion coefficient \(D(\rho)=\rho^n\), for \(n\in\mathbb{N}_+\), into a gradient model associated with the diffusion coefficient \(D(\rho)=\rho^n(1-\rho)^k\) for any \(n,k\in\mathbb{N}\). I am also going to discuss some applications of this model and open problems, and, if time allows, address its long-range extension.
  6. Gabriela Corrêa Gonçalves (UFJF),
    Estratégias de Otimização para Tratamento de Câncer com Células CAR-T: Uma Abordagem Matemática e Estatística TBA.
  7. Grégoire Véchambre (AMSS, Chinese Academy of Sciences),
    Spectral analysis of a class of Lévy-type processes and connection with some spin systems We consider a class of Lévy-type processes on which spectral analysis technics can be made to produce optimal results, in particular for the decay rate of their survival probability and for the spectral gap of their ground state transform. This class is defined by killed symmetric Lévy processes under general random time-changes satisfying some integrability assumptions. Our results reveal a connection between those processes and a family of spin systems. This connection, where the free energy and correlations play an essential role, is, up to our knowledge, new, and relates some key properties of objects from the two families. When the underlying Lévy process is a Brownian motion, the associated spin system turns out to have interactions of a rather nice form that are a natural alternative to the quadratic interactions from the lattice Gaussian Free Field. More general Lévy processes give rise to more general interactions in the associated spin systems.
  8. Gustavo Oshiro de Carvalho (IME-USP),
    The frog model with random survival parameter We consider a system of interacting random walks known as the frog model on \(\mathbb{Z}\). Initially, a random number of particles are placed at each vertex of \(\mathbb{Z}\). The particles initially at zero are active while all others are inactive. At each instant of time, each active particle may die with probability \(1-p\). Every active particle performs a simple symmetric random walk on \(\mathbb{Z}\) until the moment it dies, activating all inactive particles it hits along its path. It is already known that, under mild conditions for the initial number of particles, this process has zero probability of survival for any parameter \(p<1\). In this talk, we show that it is possible to have a positive probability of survival as well as recurrence by allowing each particle to have its own parameter p randomly selected by i.i.d variables with Beta distribution with specific parameters. For some other parameters of the Beta distribution, we show that survival has probability zero. Joint work with Fábio Machado.
  9. Hidde van Wiechen (Delft University of Technology),
    Large Deviations of the Multi-Species Symmetric Exclusion Process In 1989 Kipnis, Olla and Varadhan proved the large deviation principle for the symmetric exclusion process by looking at the hydrodynamic limit of the weakly-asymmetric simple exclusion prcess. In this joint work with Francesco Casini and Frank Redig, we look at the large deviations of the symmetric exclusion process with three types of particles. By adding a new type of particle, there is a second external field appearing in the weakly-asymmetric exclusion process and we end up with coupled equations in the hydrodynamic limit.
  10. Jhon Kevin Astoquillca Aguilar (University of Groningen),
    The voter model on dynamic random environments The voter model is an interacting particle system describing the collective behavior of voters who constantly update their political opinions on a given graph. Following a graphical representation argument, we see that this Markov process is dual of a system of coalescing random walks on the graph. This duality relationship implies that the characterization of the set of stationary measures of the voter model is linked to the dynamics of the collision of random walks on graphs.

Thursday 10h15

  1. João Luiz de Oliveira Madeira (University of Bath),
    Can deleterious mutations surf deterministic population waves? - Functional law of large numbers for a spatial model of Muller's ratchet In this work, we study the deterministic scaling limit of a model introduced by Foutel-Rodier and Etheridge to study the impact of cooperation and competition on the fitness of an expanding asexual population whose individual birth and death rates depend on the local population density. The interacting particle system can be mathematically described as particles performing symmetric random walks that undergo a birth-death process with rates that depend on the local number of particles. Phenomenologically, each particle represents a chromosome, and we keep track during the process of two features of each particle: its location and its number of deleterious mutations. After each birth event, with some positive probability, the daughter particle can acquire an additional mutation which will decrease its reproduction rate when compared to its parent. We show that under an appropriate scaling, the process converges weakly to an infinite system of partial differential equations, proving a conjecture of Foutel-Rodier and Etheridge. For the case where the reaction term satisfies a Fisher-KPP condition, we prove a conjecture of Foutel-Rodier and Etheridge regarding the spreading speed of the population into an empty habitat. We also prove some further results regarding the asymptotic behaviour of the system of PDEs in the monostable case.
  2. João Pedro Loewe Mangi (IMPA),
    Exclusion Processes with Non-Reversible Boundary: Hydrodynamics and Large Deviations We consider a family of exclusion processes defined on the discrete interval with weak boundary interaction that allows the creation and annihilation of particles on a neighborhood of radius \(L\) of the boundary under very general rates. We prove that the hydrodynamic equation is the heat equation with non-linear Robin boundary conditions. We present a particular choice of boundary rates for which we have multiple stationary solutions but for which it still holds the uniqueness of the solution of its hydrodynamic equation. We also prove the associated dynamical large deviations principle. Joint work with Beatriz Salvador and Claudio Landim.
  3. João Vitor Teixeira Maia (Peking University),
    Phase Transitions in Multidimensional Long-Range Random Field Ising Models In this talk we study the semi-infinite Ising model with an external field \(h_i = \lambda |i_d|^{-\delta}\), \(\lambda\) is the wall influence, and \(\delta>0\). This external field decays as it gets further away from the wall. The Random Cluster representation for the Ising model allow us to prove that, when \(\delta<1\), we have only one Gibbs state for any positive \(\beta\) and \(\lambda\). In addition, when \(\delta>1\) and \(\beta > \beta_c(d)\), there exists a critical value \(0< \lambda_c:=\lambda_c(\delta,\beta)\) such that, for \(\lambda<\lambda_c\) there is phase transition and for \(\lambda>\lambda_c\) we have uniqueness of the Gibbs state.
  4. Joedson de Jesus Santana (UFBA),
    A strong large deviation principle for the empirical measure for random walks Let \(X=(X_t)_{t\geq 0}\) be a Markov process on a Polish space \((\Sigma, d)\). In the 1970's, Donsker and Varadhan showed that under suitable conditions the empirical measure defined by \[L_t = \frac{1}{t} \int_0^t 1_{X_s} ds \] satisfies a large deviation principle in the space of probability measures equipped with the weak topology given that \(\Sigma\) is compact. When the state space \(\Sigma\) is not compact, the upper bound holds for \emph{compact} sets rather than closed sets. n this work, we show that under certain conditions, certain continuous-time random walks satisfy a strong large deviation principle with respect to a topology introduced by Mukherjee and Varadhan in 2016, which is natural in models exhibiting invariance with respect to spatial translations. This work applies, in particular, to the case of simple random walks and complements the results obtained by Mukherjee and Varadhan in 2016, in which the large deviation principle was established for the empirical measure of Brownian motion.
  5. Jonathan Junne (TU Delft)
    Quantitative Concentration Inequalities for Empirical Measures and Graph Laplacians on Riemannian Manifolds: An Application to the Symmetric Exclusion Process We define the symmetric exclusion process (SEP) on random neighbourhood graphs drawn from Poisson point processes on a complete connected Riemannian manifold equipped with a Gibbs reference measure \(\mu = e^{-U}d\text{Vol}\) and prove the hydrodynamic limit of the (SEP) under a curvature-dimension condition CD(\(K\),\(\infty\)) as a byproduct of quantitative concentration inequalities that we establish both for empirical measures under Ricci curvature lower bound and for graph Laplacians. We also lift the (SEP) to principal bundles, amongst which the orthonormal frame bundle plays a central role in the Eells-Elworthy-Émery construction of Riemannian Brownian motion, and we deduce horizontal diffusions as hydrodynamic limits.
  6. Julian Alexandre de Amorim (IMPA),
    Quantitative Hydrodynamics For A Generalized Contact Model We derive a quantitative version of the hydrodynamic limit obtained for an interacting particle system inspired by integrate-and-fire neuron models. More precisely, we show that the \(L^{2}\)-speed of convergence of the empirical density of states in a generalized contact process defined over a d-dimensional torus of size \(n\) is of the optimal order \(\mathcal{O}(n^{d/2})\). In addition, we show that the typical fluctuations around the aforementioned hydrodynamic limit are Gaussian, and governed by an inhomogeneous stochastic linear equation.
  7. Leonardo Andrés Videla Muñoz (Universidad de Santiago),
    Simplified Wright-Fisher processes and their non-linear extensions We introduce a two-parameter family of elementary Markov kernels on the unit interval which, under appropriate scaling, are shown to converge in distribution to Wright--Fisher diffusions with drift; a large class of drifts modeling mutation and/or selection can be recovered by playing around with the scaling. Moment dualities are obtained at the discrete level for some combinations of parameters, giving rise to discrete-time line-counting processes. This unified, parsimonious derivation of some well-known population genetic processes allows one to easily extend the dynamics to the case where mutation and/or selection rates may depend on the instantaneous distribution of the process. In this setting, we prove both, the propagation of chaos for the associated particle system in the mean-field regime, and the convergence of the rescaled discrete-time non-linear Wright-Fisher conglomerate to McKean-Vlasov-type diffusions. As an important intermediate step, it is proved that the sequence of (laws of) some non-linear Markov chains converges to the unique solution of a non-linear martingale problem.
  8. Lucas Affonso (IME-USP),
    Decay of correlations for the Long range Ising model The analysis of correlations is a central problem in statistical mechanics. For ferromagnetic long range Ising spin systems, it was proved in the 1970's by Soulliard and Iagolnitzer that the two-point correlation function cannot decay faster than the interaction strength. This implies that long-range systems with \(J_{x,y} = |x-y|^{-\alpha}\) the decay is at least algebraic. In this seminar, we discuss the current status of understanding of the decay of correlations for long-range Ising models with algebraic decay, and our result using a new cluster expansion for contours better suited to the long-range nature of the interaction. We show that, at low enough temperatures, the correlations decay algebraically also with the same exponent as the interaction.
  9. Lucas Roberto de Lima (UFABC),
    Asymptotic Shape of Subadditive Processes on Groups with Polynomial Growth This presentation delves into the investigation of the limiting shape theorem for subadditive processes on finitely generated groups that exhibit polynomial growth, also known as virtually nilpotent groups. We explore the underlying algebraic structures of these processes, offering a generalization of the asymptotic shape theorem within this context. We extend the subadditive ergodic theory in this setting to consider processes that exhibit at least and at most linear random growth. We conclude with applications and illustrative examples. This is a joint work with Cristian F. Coletti, preprint available on arXiv:2312.07811.
  10. Luciano Henrique Lacerda de Araújo (UFABC),
    Asymptotic Shape of Subadditive Processes on Groups with Polynomial Growth Let \(G = (V, E, \phi)\) be an infinite, with uniformly bounded degree, connected, and weighted graph, where \(\phi\) is its weight function. Each edge is associated with a Bernoulli random variable and is called open with probability \(p\) and closed with probability \(1 - p\), where \(0 < p < 1\). Fixing a vertex, which will be called the origin of the graph, we will consider the box \(G[B_n]\) as the subgraph of \(G\) whose vertices are those whose distances from the origin are equal to or less than \(n\), and whose edges are the edges of \(G\) between vertices of \(G[B_n]\). A simplicial complex on this subgraph is a collection of subsets of its vertices where, for any set in the collection, all of its subsets are in the collection too. A set in the collection with \(k + 1\) vertices is called a \(k\)-simplex, and we will consider the vector space generated by all the \(k\)-simplex of the simplicial complex, which is called the vector space of the \(k\)-chains of \(G[B_n]\). Our focus is the local simplicial complexes, which are characterized by the existence of a positive constant \(R\), such that for each simplex, each pair of vertices in this simplex are distant by no more than \(R\). By building a sequence of linear operators called boundary operators we can define the homology groups of the subgraph \(G[B_n]\), whose basic elements are called \(k\)-generators of homology. It occurs that, if we remove the closed edges, the list of \(k\)-simplices, the \(k\)-chains and the dimensions of the homology groups, called Betti numbers, and denoted \(\beta^n_k\), depend on the randomness of the graph, which means that the Betti numbers are random variables. Our aim is to understand the desirable additional properties for the graph that guarantee, and prove that, if the graph has infinite independent \(k\)-generators of homology with the same probability of occuring, than the sequency of normalized Betti numbers, as we enlarge the box \(G[B_n]\), satisfies the Central Limit Theorem. As a techinique to reach our goals, we are decomposing the Betti numbers into sums of differences of martingales and using the tools from the Martingale Theory.

Thursday 16h

  1. Manuel Astiazarán Moreira (Universidad de la República),
    Graph Algorithms Applied to Identity Resolution in a Real Flights Dataset Disambiguating different registers corresponding to the same person is a complex problem with multiple approaches, and it is of great interest in various areas, from commercial and marketing perspectives to security and state information systems. Recently, several authors have proposed using graph techniques in identity resolution to improve the performance of traditional models. The graph formulation allows the incorporation of information from both the attributes of each record and their structural relationships. In this joint work with Mauricio Genta, we apply the Louvain Community Detection Algorithm (LCDA) to a flights dataset, where each node represents a record. We explore different methods of establishing edges based on attributes and relationships between the records, aiming to identify real-life individuals as connected components in the resulting graph from the LCDA.
  2. Marcel Schrocke (Universität Augsburg),
    Compass Model The Compass model explores opinion dynamics within a population where agents adjust their opinions (modelled by points on a circle) through pairwise interactions. When the difference between two interacting agents' opinions is below a threshold \(\theta\), they adjust their opinions towards each other. We investigate how various phenomena such as consensus formation, opinion clustering, and polarization depend on the value of \(\theta\).
  3. Marco Antonio Ticse Aucahuasi (UFMG),
    A Review of Expectation: The Case of the Inverse of the Binomial Random Variable TBA.
  4. Mariana Pereira Lopes (UFMG),
    Point Processes Extreme value theory concerns modelling situations where we want to predict the maximum (or minimum) of random values. Examples include predicting the maximum height that a river will reach in order to construct a dam, the amount of pressure endured by a plane wing during flights or the breaking of Olympic records. Mathematically, (one-dimensional) extreme value theory examines the convergence in distribution of the values \(V_{1\leq i\leq n} X_n = \max_{1\leq i\leq n} X_i\) for \((X_n)_{n\in \mathbb N}\) independent and identically distributed random variables with a common distribution function \(F\). In this investigation, an important role is played by regularly varying functions: measurable functions \(U : \mathbb R_+ \to \mathbb R_+\) such that \(\lim_{t\to+\infty} U(tx)/U(t)= x^\rho\) for all \(x \in \mathbb R\), where \(\rho\in \mathbb R\) is a parameter. In this work, we present an important result that characterizes the behavior of extreme values, describing three possible classes of limit distributions for the maximum. We will also discuss simple sufficient conditions for the existence of \(a_n > 0\) and \(b_n \in \mathbb R\) such that \(x \mapsto F^n(a_nx + b_n)\) converges to one of the extreme value distributions, that is, for \(F\) to be in the domain of attraction of one of these functions.
  5. Rodrigo Lambert (UFU),
    A complete characterization of a correlated Bernoulli process We present a complete characterization of the asymptotic behaviour of a correlated Bernoulli sequence on the \(0-1\) alphabet. A martingale theory based approach will allow us to prove versions of the law of large numbers, quadratic strong law, law of iterated logarithm, almost sure central limit theorem and functional central limit theorem. This is joint with M. González-Navarrete (UFRo - Chile) and V. Vázquez Guevara (BUAP - Mexico).
  6. Matheus Lopes Coelho (UFMG),
    Percolação na rede triangular: expoentes críticos para comprimentos característicos Kesten's results show that it is possible to relate near-critical and critical percolation in the two dimensional triangular lattice. This allows for the derivation of critical exponents for some characteristic lengths, as presented by Nolin, which describe near-critical percolation in this model. We present some of the main tools and properties used to relate critical and near-critical percolation, such as separation lemmas, the asymptotic equivalence of near-critical arm events and the derivation of the critical exponents for characteristic lengths.
  7. Nandan Malhotra (Leiden University),
    Limiting Spectra of inhomogeneous random graphs I will describe some results on the bulk of the spectrum of the adjacency matrices of sparse and dense inhomogeneous random graphs, and show how the two spectrums are related. Based on joint work with Luca Avena and Rajat Subhra Hazra.
  8. Pablo A. Gomes (IME-USP),
    Truncated long-range percolation of words We consider a model for percolation of words in a random lattice. The random lattice is obtained by an independent percolation process on the edges of the hypercubic lattice with long-range edges in one of the directions. Given a sequence of parameters \(p_1,p_2,...\), \(0 \leq p_n \leq 1\), an edge of length \(n\) is declared open with probability \(p_n\). We prove that if the sum of terms of the sequence of parameters diverges, then, almost surely, there is a \(K\) large enough such that all words are seen from the origin simultaneously, even if all connections larger than \(K\) are suppressed. This is a joint work with Otávio Lima and Roger W. C. Silva.
  9. Pablo Ignacio Araya Zambra (Universidad de Chile),
    Theory of Explorable Sets In order to study the level sets of the Vectorial Gaussian Free Field (GFF), it's necessary to overcome two problematics. The GFF is not a function, and the intersection of level sets is not Markovian in general. To address this, we propose a family of Markovian sets that overcome these problems, called "Explorable sets". These sets can be thought of as a type of set that can be discovered from the boundary in an adapted way. In this talk, we present this family, along with some properties and examples in various models. In particular, we show that two types of Markovian sets associated with the GFF (FPS, TVS) are explorable sets, thanks to a limit result of the explorable sets. Finally, we mention the non-existence of the TVS and FPS in higher dimensions using this family of sets.
  10. Paul Martinez Vilca (UnB),
    Construção Alternativa do Movimento Browniano TBA.

Friday 10h15

  1. Paul Nikolaev (Universität Mannheim),
    Relative entropy estimates for convolution interaction forces Quantitative estimates are derived, on the whole space, for the relative entropy between the joint law of random interacting particles and the tensorized law at the limiting system. The developed method combines the relative entropy method under the moderated interaction scaling introduced by Oeschläger, and the propagation of chaos in probability. The result includes the case that the interaction force does not need to be a potential field. Furthermore, if the interaction force is a potential field, with a convolutional structure, then the developed estimate also provides the modulated energy estimates. Moreover, we demonstrate propagation of chaos for large stochastic systems of interacting particles and discuss possible applications to various interacting particle systems, including the Coulomb interaction case.
  2. Rebeca Alves Dantas de Lima e Silva (UFMG),
    Inhomogeneous time diffusion processes as solutions for SDE's Exact simulation methods are applicable to diffusion processes with constant behavior over time, known as homogeneous time, as cited by Alexandros Beskos, Omiros Papaspiliopoulos, and Gareth O. Roberts. However, in the current literature, it is briefly mentioned that all these methods can be extended to diffusion processes in which the characteristics of the process, such as drift and diffusion coefficient, vary over time, referred to as inhomogeneous time. Thus, the aim of this poster is to explore exact simulation for inhomogeneous time diffusion processes. These processes are the solution of SDEs of the form: \(dY_s = b(s, Y_s; \theta)ds + \sigma(s, Y_s; \theta)dB_s\), \(Y_0 = y \in \mathbb R, s \in [0, T]\). In this equation, \(b(s, Y_s; \theta)\) is the drift, \(\sigma(s, Y_s; \theta)\) is the diffusion coefficient, \(B_s\) is a standard Brownian motion, \(\theta \in \Theta\). The objective is to study methods for simulating trajectories of this type of process.
  3. Roger William Câmara Silva (UFMG),
    Constrained degree percolation in random environment We consider the Constrained-degree percolation model in random environment on the square lattice. In this model, each vertex \(v\) has an independent random constraint \(\kappa_v\) which takes the value \(j \in \{0, 1, 2, 3\}\) with probability \(\rho_j\). Each edge \(e\) attempts to open at a random uniform time \(U_e \in [0, 1]\), independently of all other edges. It succeeds if at time \(U_e\) both its end-vertices have degrees strictly smaller than their respectively attached constraints. We show that this model undergoes a non-trivial phase transition when \(\rho_3\) is sufficiently large. The proof consists of a decoupling inequality, the continuity of the probability for local events, and a coarse-graining argument.
  4. Ruan dos Santos Vieira Miranda (IME-USP),
    Moderates and Consensus Formation in the Deffuant Model on the Square Lattice Understanding the conditions for consensus formation is a pivotal topic in social, behavioral, economic, and other sciences. Opinion dynamics, particularly in the mathematical sense, provides valuable insights into this phenomenon. The Deffuant model describes consensus formation where pairs of agents interact only if their opinions differ by no more than a given threshold, leading to an approximation of their beliefs. Traditionally, this model is defined on a graph \(G=(\mathbb{Z},E)\) with \(E=\{(x,x+1):x\in \mathbb{Z}\}\) and initial opinions uniformly distributed on the unit interval. In our research, we extend the model to \(\mathbb{Z}^2\) and explore various initial distributions to assess the role of moderates — agents with opinions inside the opinion spectrum —, in addition to the extremists, in achieving consensus. We aim to determine the critical quantity of moderates necessary for consensus formation. Numerical evidence suggests that in populations with a high tolerance to interact, even a small number of moderates can significantly impact the speed of achieving consensus.
  5. Sandro Gallo (UFSCar),
    Sub-Gaussian concentration inequalities for random fields Since McDiarmid inequality who obtained sub-gaussian concentration for some general classes of functionals of finite samples of independent random variables, the literature has tried to extend these results to other classes of models. For instance, Kulske (2003) obtained such inequalities for random fields satisfying some Dobrushin like assumption, whereas Chazottes, Collet, Kulske and Redig (2006) proved that for sufficiently high temperature, the ferromagnetic Ising model on \(\mathbb Z^d\) satisfies sub-Gaussian concentration. In a work in progress with Jean-René Chazottes and Daniel Y. Takahashi, we proved that sub-gaussian concentration holds as long as we are above the critical temperature in the ferromagnetic Ising model.
  6. Sebastian Angel Zaninovich (Universidad de Buenos Aires),
    Continuity of the time constant in Euclidean First-Passage Percolation and Fermat distances in the presence of Noise In this work, we introduce a model of Euclidean First Passage Percolation in the presence of noise. We establish continuity results as the noise converges to zero and provide bounds related to the non-noisy time constant. Additionally, we apply the same techniques to prove analogous results for the Fermat distance - a metric defined on datasets that infers, in some sense, the geometry of the manifold where the data is supported. Joint work with Pablo Groisman.
  7. Seoyeon Yang (Princeton University),
    Cut-off and dynamical phase transition for the general multi-component Ising model We study the multi-component Ising model, which is also known as the block Ising model. In this model, the particles are partitioned into a fixed number of groups with a fixed proportion, and the interaction strength is determined by the group to which each particle belongs. We demonstrate that the Glauber dynamics on our model exhibits the cutoff-metastability phase transition as passing the critical inverse-temperature \(\beta_{cr}\), which is determined by the proportion of the groups and their interaction strengths, regardless of the total number of particles. For \(\beta<\beta_{cr}\), the dynamics shows a cutoff at \(\alpha n\log n\) with a window size \(O(n)\), where \(\alpha\) is a constant independent of \(n\). For \(\beta=\beta_{cr}\), we prove that the mixing time is of order \(n^{3/2}\). In particular, we deduce the so-called non-central limit theorem for the block magnetizations to validate the optimal bound at \(\beta=\beta_{cr}\). For \(\beta>\beta_{cr}\), we examine the metastability, which refers to the exponential mixing time. Our results, based on the position of the employed Ising model on the complete multipartite graph, generalize the results of previous versions of the model.
  8. Sérgio de Carvalho Bezerra (UFPB),
    Computation Results for a site percolation model in \(\mathbb{Z}^2\) with two fluids and local dependence In this poster, we show some results of site percolation with two fluids (red and blue). A local dependence appears over the site percolation with two fluids, otherwise we are in the same case of only one fluid. The environment of our simulation is a square centered in the origin. The spiral walk along the sites of the square was chosen. At each site there are four possible directions to proceed. For each direction, one of \(3\) possibilities may occur: a blue site with probability \(q\), two red sites with probability \(p\) and nothing with probability \(1-p-q\). Our main result is to obtain numerically the value \(p_c = \inf p\) such that with probability \(1\) there is no blue fluid walk between the east and west sides of the square.
  9. Shangjie Yang (IME-USP),
    Mixing time for the asymmetric simple exclusion process in a random environment We consider the simple exclusion process in a finite line segment where the jumping rates of particles are independently sampled from a common law, which is such that a random walk on the whole line is transient. We prove polynomial lower and upper bounds on the mixing time. Joint work with Hubert Lacoin (IMPA).
  10. Stefan Nicolae Paicu (TU Delft),
    Large deviations: from the discrete case to stochastic differential equations with small noise diffusions I will talk about the general framework of large deviations theory and present the main results in the discrete case. Then, its generalization to stochastic processes, and in particular to Brownian motion, is analyzed. Finally, a large deviations principle for a class of stochastic differential equations with small noise diffusions is stated and proved with two different techniques. Based on my master thesis supervised by Prof. Carlo Orrieri.

Friday 16h

  1. Vicente Lenz (TU Delft),
    Metastability for the dilute Curie-Weiss Potts model We analyse the metastable behaviour of the disordered Curie-Weiss-Potts (DCWP) model subject to a Glauber dynamics. The model is a randomly disordered version of the mean-field q-spin Potts model (CWP), where the interaction coefficients between spins are general independent random variables. This model comprises also, e.g., the CWP model on inhomogeneous dense random graphs. We are interested in a comparison of the metastable behaviour of the CWP and the DCWP models, for fixed temperature and the infinite volume limit. We prove the CWP model is metastable and through this prove metastability for the DCWP model. Then we identify the ratio of the (random) mean time the DCWP model takes to reach the stable phase when it starts from a certain probability distribution on the metastable state (called the last-exit biased distribution) and the (non-random) corresponding quantity for the CWP model. In particular, we obtain the asymptotic tail behaviour and the moments of the ratio of the metastable hitting times of the disordered and annealed systems. Our proof is based on a combination of the potential theoretic approach to metastability and concentration of measure techniques, the later adapted to our particular setup.
  2. Vicenzo Bonasorte Reis Pereira (UFSCar),
    Rumor Spreading in Dynamic Random Graphs We study rumor spreading in dynamic random graphs. Starting with a single informed vertex, the information flows until it reaches all the vertices of the graph (completion), according to the following process. At each step k, the information is propagated to neighbours, in the k-th generated random graph,of the informed vertices. The way this information propagates from vertex to vertex at each step will depend of the "protocol". First, we consider the rumour spreading according to the Push Protocol (at every round, informed nodes send the rumour to one of their neighbours, chosen uniformly at random) in a sequence of independent stochastic block model random graphs. We are able to bound the completion time in this setting using comparisons with rumour spreading in dynamic random graphs with skeptical nodes (nodes that cannot become informed) and stifler nodes (nodes that, after being informed, do not spread the information further). We also consider a sequence of graphs in which the presence or absence of an edge follows the dynamic of a Markov chain. We provide a method based on strong stationary times allowing to bound the completion time for the Markov dynamic using bounds on the completion time in the i.i.d. dynamic.
  3. Vinicio Rodrigues Moreira (UFMG),
    An introduction to the problem of Large Loss in Credit Risk Portfolios In this poster we are interested in the study of events that are rare but, when they happen, lead to big financial losses for a financial institution. These are events where an unexpectedly large number of debtors fail to pay their debts by the due date. We will focus our attention on two specific cases: in the first one, debtors are all independent, and in the second one there is a global dependence between them. Our objective is to determine the asymptotic probability of such losses in both cases when the number of debtors is large. These two cases are simple but are good starting points for the study of Large Loss in Credit Risk Portfolios.
  4. Yanyan Hu (TU Delft),
    Large deviations for Cox-Ingersoll-Ross processes with state-dependent fast switching We study the large deviations for Cox-Ingersoll-Ross (CIR) processes with small noise and state-dependent fast switching via associated Hamilton-Jacobi equations. As the separation of time scales, when the noise goes to 0 and the rate of switching goes to \(\infty\), we get a limit equation characterized by the averaging principle. Moreover, we prove the large deviation principle (LDP) with an action-integral form rate function to describe the asymptotic behavior of such systems. The new ingredient is establishing the comparison principle in the singular context. The proof is carried out using the nonlinear semigroup method coming from Feng and Kurtz's book.
  5. Zoraida Rico (Columbia University),
    Fine bounds on statistical estimation In this talk, we will discuss optimal statistical estimation for finite samples from the perspectives of robustness and heavy-tailed data. We study two problems: properties of the trimmed mean, and covariance matrix estimation.