Crossing probabilities in geometric inhomogeneous random graphs 

Amitai Linker (Universidad Andrés Bello, Chile)

Resumo: In geometric inhomogeneous random graphs, vertices are given by the points of a Poisson process on $\mathbb{R}^d$ and are equipped with independent weights with a heavy tailed distribution. Any pair of distinct vertices independently forms an edge with a probability decaying as a function of the product of their weights divided by their distance. Recent work of Jacob et al. provides criteria for existence of a quantitatively subcritical phase in this model, that is, a phase where the probability of crossing an annulus with proportional inner and outer radii tends to zero as the radii tend to infinity. In our work we consider more general annuli by relaxing the proportionality assumption, and show that in the quantitatively subcritical phase the crossing probabilities are equivalent to the crossing probabilities using one or two edges. Moreover, we give the explicit asymptotic order of these probabilities (and hence, of the crossing probability), which depends on the inner and outer radius of the annulus, the power-law exponent of the degree distribution, and the decay of the probability of long edges. As a corollary we get the subcritical one-arm exponents characterizing the decay of the probability that a typical point is in a component not contained in a centred ball whose radius goes to infinity. Based on joint work with Emmanuel Jacob, Céline Kerriou and Peter Mörters. 

Oriented percolation with inhomogeneities and strict inequalities 

Bernardo Lima (UFMG)

Resumo: This work was motivated by natural questions related to oriented percolation on a layered environment that introduces long range dependence. 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 D. Ungaretti and M. E. Vares.

Fluctuations of the occupation density for a parking process

Cristian Coletti (UFABC)

Resumo: Consider the following simple parking process on $\Lambda_n=\{-n,\ldots,n\}^d, d \geq 1$: at each step, a site  is chosen at random in $\Lambda_n$ and if  and all its nearest neighbor sites are empty,  is occupied. Once occupied, a site remains so forever. The process continues until all sites in  are either occupied or have at least one of their nearest neighbors occupied. The final configuration (occupancy) of $\Lambda_n$ is called the jamming limit and is denoted by $X_{\Lambda_n}$. Ritchie (2006) constructed a stationary random field on $\mathbb{Z}^d$ obtained as a (thermodynamic) limit of the $X_{\Lambda_n}$'s as $n$ tends to infinity. As a consequence of his construction, he proved a strong law of large numbers for the proportion of occupied sites in the box $\Lambda_n$  for the random field $X$. Here we prove the central limit theorem, the law of iterated logarithm, and a gaussian concentration inequality for the same statistics. A particular attention will be given to the case $d=1$. 

Finitary Codings and Gaussian Concentration for Random Fields 

Daniel Takahashi (UFRN)

Resumo: We prove that finite-valued shift-invariant random fields on Z^d that are finitary codings of an i.i.d. random field satisfy a Gaussian concentration bound (GCB), provided that the coding volume has a finite second moment. As an application, we significantly strengthen several existing results on GCB for lattice models. For instance, we establish a necessary and sufficient condition for GCB to hold in the ferromagnetic Ising model for all d ≥ 2. Our approach provides an alternative and constructive proof even when the results are known. This is a joint work with J.-R. Chazottes and S. Gallo.

Particle systems on top of random graphs 

Guilherme Reis (UFF)

Resumo: In this talk, we will discuss particle systems defined on sequences of random graphs, with a particular focus on interacting diffusions. The sequences of random graphs we consider include sequences of Erdős–Rényi graphs with different levels of sparsity. We will also present some recent results and highlight open questions related to the voter model. 

Limit Laws for Equilibrium States in Dynamical Systems 

Jaqueline Siqueira (UFRJ)

Resumo: In general, a dynamical system may admit multiple invariant probability measures. A natural and effective approach to selecting a meaningful measure is to focus on those that maximize the system's topological pressure—these are referred to as equilibrium states. This talk explores the uniqueness of equilibrium states for certain classes of non-uniformly hyperbolic systems. Additionally, it addresses limit laws such as the central limit theorem and the exponential decay of correlations. (Based on several joint works with  José Alves, Suzete Afonso, Stefano Luzzato and Vanessa Ramos). 

On the One-Dimensional Contact Process with Enhancements 

Leonardo Rolla (USP)

Resumo: We study a one-dimensional contact process with two infection parameters, one giving the infection rates at the boundaries of a finite infected region and the other one the rates within that region. We prove that the critical value of each of these parameters is a strictly monotone continuous function of the other parameter. We also show that if one of these parameters is equal to the critical value of the standard contact process and the other parameter is strictly larger, then the infection starting from a single point has a positive probability of surviving. This is in contrast with another result also obtained here, that the critical contact process on the half line with enhanced infection rate at finitely many sites also dies out. Joint work with E. Andjel. 

The Truncation Question in Percolation of Words 

Pablo Gomes (USP) 

Resumo: In this talk, we investigate the truncation question in the context of percolation of words. Consider the set Z^d, where each vertex is independently assigned the letter 1 with probability p, and the letter 0 with probability 1-p. Long-range edges of length n are declared open with probability p_n. The goal is to analyze the existence of a sufficiently large K such that, with positive probability, all words are seen from the origin, even if the edges of length larger than K are removed from the lattice. Based on joint work with Otavio Lima and Roger W. C. Silva. 

Survival of one-dimensional Renewal Contact Process (SLIDES)

Rafael Souza dos Santos (UFRJ)  

Resumo: The renewal contact process, introduced in 2019 by Fontes, Marchetti, Mountford, and Vares, extends the Harris contact process by allowing the possible cure times to be determined according to independent renewal processes with some interarrival distribution \mu and keeping the transmission times determined according to independent exponential random variables with a fixed rate \lambda. In this talk we will discuss sufficient conditions on \mu to have a positive and finite critical parameter in the renewal contact process. Joint work with Maria Eulalia Vares.  

Weakly self-avoiding random walk in a random potential 

Renato Soares dos Santos (UFMG)

Resumo: We consider a model of a simple random walk influenced by two competing types of interaction: an attractive one towards high values of a random potential, and a self-repellent one measured by its self-intersections (in the spirit of the weakly self-avoiding random walk). We identify the log asymptotics of the partition function of the model and also the typical path behaviour giving the main contribution to the partition function. The latter comes out of a variational formula and shows concentration on a random finite number of points, each occupied for a positive fraction of time. Joint work with Wolfgang König, Nicolas Pétrélis and Willem van Zuijlen. 

Sprinkling for the Hammersley Process and Applications 

Roberto Viveros (UFRJ)

Resumo: I will present recent results on a decoupling inequality with sprinkling for the Hammersley process, a totally asymmetric interacting particle system. Based on this inequality, I will discuss two applications: a detection problem, in which a target seeks to avoid mobile detectors, and a ballisticity result for a random walk whose dynamics are governed by the environment. 

The Tree-Builder Random Walk: Speed and Structure 

Rodrigo Ribeiro (University of Denver)

Resumo: The Tree-Builder Random Walk (TBRW) is a dynamic process where a particle explores a set of vertices while simultaneously adding new ones, building a growing random tree. The model bridges the theories of random walks and evolving random graphs. This talk introduces the TBRW and discusses its two main perspectives. We will then focus on two central objects: the speed of the random walk and the graph topology of the resulting random tree (structure). The presentation is based on a solo work by the speaker and a joint work with Janos Englander (University of Colorado), Giullio Iacobelli (UFRJ), and Gábor Pete (Alfred-Renyi Institute of Mathematics). 

Spectral Gap and Cutoff for the Simple Exclusion Process with IID Conductances 

Shangjie Yang (UFF)

Resumo: In this talk, we study the spectral gap and the mixing time of the simple exclusion process on a finite linear segment, where the contents of two sites $x, x+1$ are swapped at rate c(x, x+1). Under certain assumptions on the c(x, x+1)_x and the number of particles, we identify the spectral gap and prove that the total variation distance to equilibrium drops abruptly from 1 to 0 at a specific time.

Limit Theorems for Random Dirichlet Series (SLIDES)

Susana Frometa (UFRGS)

Resumo: We consider a random Dirichlet series $F(\sigma) = \sum_{n=1}^\infty\frac{X_n}{n^\sigma}$ where $(X_n)_{n \in \mathbb N}$ are independent and identically distributed random variables. We assume that $\mathbb E(X_n)=0$ and $\mathbb E(X_n^2)<\infty$, so the series converges almost surely for $\sigma>1/2$. We are interested in the behavior of $F(\sigma)$ as $\sigma$ approaches $1/2$. In this talk, we will present two main results. First, for the Rademacher case, we provide a precise description of the magnitude of the oscillations of $F(\sigma)$ as $\sigma\to1/2^+$, by showing that it satisfies a Law of the Iterated Logarithm. In particular, this implies that, almost surely, $F(\sigma)$ has an infinitely many real zeros accumulating at $\sigma=1/2$. Then, in the standard Gaussian case, we obtain a quantitative estimate of the asymptotic average number of real zeros of $F(\sigma)$ in the interval $(\tau,+\infty)$, as $\tau\to1/2^+$. This is a joint work with Marco Aymone and Ricardo Misturini.