In 1982, Fröhlich and Spencer solved a long-standing conjecture about phase transition in a one-dimensional long-range Ising model for 1/r^2 interaction energy. After that, Cassandro, Ferrari, Merola, and Presutti extended the contour argument for other decays alpha. In this talk, we address the multidimensional long-range Ising model, showing that a contour argument holds for all decays alpha> d. As an application, we will show how our techniques can be used to study phase transition when the model has a decaying magnetic field.

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

The Ant RW belongs to the class of random walks with reinforcement. Our goal is to observe through this stochastic process the ant mill phenomenon: the walker is eventually trapped in a circuit of the graph which will be followed forever. In recent work we show that this phenomenon can be observed with weaker assumptions in the reinforcement function. Ours inspiration is the ant mill phenomenon in which a group of army ants are separated from the main group, lose the pheromone track and begin to follow one another, forming a continuously rotating circle. The ants are not able to go back home and will eventually die of exhaustion as we can see in the video “Why army ants get trapped in ‘death circles’” on youtube: https://www.youtube.com/watch?v=LEKwQxO4EZU

Works in collaboration with Dirk Erhard and Tertuliano Franco.

Wilson’s algorithm efficiently samples spanning forests of a given network that are associated with a partition function that coincides, in accordance with a theorem by Kirchhoff, with the characteristic polynomial of the infinitesimal generator of the continuous time random walk on the network. This provides a probabilistic proof of this theorem and we will discuss how it also gives access to various Markov spectrum properties and estimates.

Abstract: We consider a system of $k$ particles on a segment of size $N$. Its rules of evolution are the following: a particle at site $x in {1 ,dots, N}$ jumps to $x+1$ with rate $omega_x$ and to $x-1$ with rate $1-omega_x$ where $omega_xin (0,1)$, and a jump is cancelled if the site is already occupied. We consider the case where $(omega_x)^N_{x=1}$ is (the fixed realization of) a sequence of IID random variables. Assuming that $mathbb E[ logfrac{1-omega_x}{omega_x}]ne 0$ (that is, transience of the random environment), we prove that this particle systems mixes fast, in the sense that the time that it requires for its distribution to get close to the equilibrium state grows like a power of $N$. We present a lower bound for the power in mixing time which we conjecture to be sharp.

Joint work with S. Yang, IMPA.

The title of the talk corresponds to a family of interesting random processes, which includes lazy random walks on graphs and much beyond them. Often, a key step in analysing such processes is to estimate their spectral gaps (ie. the difference between two largest eigenvalues). It is thus of interest to understand what else about the chain we can know from the spectral gap. We will present a simple comparison idea that often gives us the best possible estimates. In particular, we re-obtain and improve upon several known results on hitting, meeting, and intersection times; return probabilities; and concentration inequalities for time averages. We then specialize to the graph setting, and obtain sharp inequalities in that setting. This talk is based on work that has been in progress for far too long with Yuval Peres.

RWRE (Random Walk in Random Environment) is a classical model for particles moving in a non-homogeneous medium presenting impurities. It consists of a random walk on a graph with random transition probabilities determined by an underlying (static) field of random variables. The long term behavior of RWRE is well-understood, at least on the one-dimensional integer lattice, where trapping effects due to the spatial non-homogeneities lead to very different results than for a standard homogeneous random walk (e.g. non-local recurrence criterion, transient sub-ballistic regimes, anomalous diffusions, sub-exponential large deviations, aging).

In this talk we are interested in perturbing the underlying static random environment by repeatedly re-sampling it from a given law along a sequence of prescribed times, the so-called cooling sequence. This perturbation makes the environment dynamic and the resulting model, recently introduced in the literature, is referred to as RWCRE (Random Walk in Cooling Random Environment).

Depending on the choice of the cooling sequence, RWCRE may present strong homogenization as for a homogeneous Random Walk, or can lead to strong trapping effects as for RWRE. A surprisingly rich palette of possible limit scenarios have been explored in a series of recent papers and ongoing works on the one-dimensional lattice. I plan to give an account of these results and related techniques. Particular emphasis will be given to fluctuations and scaling limits where crossovers and mixtures of different laws emerge as a function of the structure of the cooling sequence.

Based on joint works with Yuki Chino (Taiwan), Conrado da Costa (Durham), Frank den Hollander (Leiden) and Jonathan Peterson (Purdue).

In this talk, we study the convergence in the strong sense, with respect to the L^2-norm, of the weak solution of the porous medium equation (for short PME) with a type of Robin boundary conditions, by tuning a parameter either to zero or to infinity. The limiting function solves the same equation with Neumann (resp. Dirichlet) boundary conditions when the parameter is taken to zero (resp. infinity).

The keystone to prove this convergence result is a sufficiently strong energy estimate to the weak solution of the PME with a type of Robin boundary conditions.
Our approach to obtaining it is to consider an underlying microscopic dynamics, given by an interacting particle system, whose space-time evolution of the density of particles is ruled by the solution of those equations. We called this microscopic dynamic by the porous medium model (PMM) with slow boundary. The relation between the PMM and PME is stated in the paper: https://link.springer.com/article/10.1007/s10955-020-02550-y, through the hydrodynamic limit for the PMM with slow boundary.

It is a joint work with Patrícia Gonçalves (IST – Lisbon) and Renato De Paula (IST – Lisbon), see more in https://arxiv.org/abs/2101.08900.

In this talk we study the existence/absence of phase transitions for Bernoulli percolation on a class of random planar graphs. More precisely, the graphs we consider have vertex sets given by Z^2 and we start by adding all horizontal edges connecting nearest neighbor vertices. This gives us a disconnected graph, composed of infinitely many copies of Z, with the trivial behavior p_c(Z) = 1. We now add to G vertical lines of edges at {X_i}xZ, where the points X_i are given by an i.i.d. integer-valued renewal process with inter arrivals distributed as T. This graph G now looks like a randomly stretched version of the nearest neighbor graph on Z^2. In this talk we show an interesting phenomenon relating the existence of phase transition for percolation on G with the moments of the variable T. Namely, if E(T^{1+eps}) is finite, then G almost surely features a non-trivial phase transition. While if E(T^{1-eps}) is infinite, then p_c(G) = 1.
This is a joint work with Hilário, Sá and Sanchis.

The Box-Ball System is a cellular automaton introduced by Takahashi and Satsuma as a discrete counterpart of the Korteweg & de Vries (KdV) differential equation. Both systems exhibit solitons, solitary waves that conserve shape and speed even after collision with other solitons. A configuration is a binary function on the integers representing boxes which may contain one ball or be empty. A carrier visits successively boxes from left to right, picking balls from occupied boxes and depositing one ball, if carried, at each visited empty box. Conservation of solitons suggests that this dynamics has many spatially-ergodic invariant measures besides the i.i.d. distribution. Building on Takahashi-Satsuma identification of solitons, we provide a soliton decomposition of the ball configurations and show that the dynamics reduces to a hierarchical translation of the components, finally obtaining an explicit recipe to construct a rich family of invariant measures. We also consider the a.s. asymptotic speed of solitons of each size. An extended version of this abstract, references, simulations, and the slides, all can be found at https://mate.dm.uba.ar/~leorolla/bbs-abstract.pdf.

This is a joint work with Pablo A. Ferrari, Chi Nguyen, Minmin Wang.

Given uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them? “Fairly” means that each region has the same area. It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting gradient flow yield such a partition-with exactly equal areas, no matter how the points are distributed. This is related to work of Nazarov-Sodin-Volberg on Gaussian analytic functions. (See the cover of the AMS Notices at http://www.ams.org/publications/journals/notices/201705/rnoti-cvr1.pdf.) Our main result is that this partition minimizes, up to a bounded factor, the average distance between points in the same cell. I will also present an application to almost optimal matching on the sphere, connecting to a classical result of Ajtai, Komlos and Tusnady (Combinatorica 1984).

Joint work with Nina Holden and Alex Zhai.

Lattice trees is a probabilistic model for random subtrees of Z^d. In this talk we are going to review some previous results about the convergence of lattice trees to the “Super-Brownian motion” in the high-dimensional setting. Then, we are going to show some new theorems which strengthen the topology of said convergence. Finally, if time permits, we will discuss the applications of these results to the study of random walks on lattice trees.

Joint work with A. Fribergh, M. Holmes and E. Perkins.

We study conditions for the existence of the asymptotic shape for subadditive processes defined on Cayley graphs of finitely generated groups with polynomial growth. We will focus our attention on the cases of First-Passage Percolation and the Frog Model. The considered class of graphs is an algebraic generalization of the hypercubic Z^d lattice, and the related limiting shape results combine probability with techniques from geometric group theory. This talk is based on a joint work with Cristian Coletti.

The Thermodynamic Formalism provides a rigorous mathematical framework for studying the quantitative and qualitative aspects of dynamical systems. At its core, there is a variational principle that corresponds, in its simplest form, to the maximum entropy principle. It has been used as a statistical inference procedure to represent the collective behavior of complex systems by specific probability measures (Gibbs measures). This framework has found applications in different domains of science. In particular, it has been fruitful and influential in neurosciences. In this talk, I will discuss and briefly review how Thermodynamic Formalism can be exploited in the field of theoretical neuroscience, as a conceptual and operational tool, to link the dynamics of interacting neurons and the statistics of action potentials from either experimental data or mathematical models. I will end this talk by commenting on perspectives and open problems in theoretical neuroscience that could be addressed within this formalism.

The task of efficiently generating uniform spanning trees of a graph has received much attention. A breakthrough came with Aldous-Broder and Wilson’s algorithms, which can efficiently generate spanning trees based on random walks. In this work, we study the transient behavior of both algorithms. We introduce the notion of branches, which are paths generated by the two algorithms on particular stopping times. This interpretation is used to show a transient equivalence between the two algorithms on complete graphs. This equivalence yields a hybrid approach to generate uniform spanning trees of complete graphs faster than either of the two algorithms. We also propose a two-stage framework to explore this hybrid approach beyond complete graphs, showing its feasibility in some examples.

We construct an infinite volume spatial random permutation associated to a Gaussian Hamiltonian, which is parametrized by the point density and the temperature. Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman (1953). Bose-Einstein condensation occurs for dimensions 3 or larger, above a critical density, and is manifest in this representation by the presence of cycles of macroscopic length. For subcritical densities we define the spatial random permutation as a Poisson process of finite unrooted loops of a random walk with Gaussian increments that we call Gaussian loop soup, analogous to the Brownian loop soup of Lawler and Werner (2004). We also construct Gaussian random interlacements, a Poisson process of doubly-infinite trajectories of random walks with Gaussian increments analogous to the Brownian random interlacements of Sznitman (2010). For dimensions greater than or equal to 3 and supercritical densities, we define the spatial permutation as the superposition of independent realizations of the Gaussian loop soup at critical density and Gaussian random interlacements at the remaining density. We show some properties of these spatial permutations, in particular that the point marginal is the boson point process, for any point density.

This is joint work with P.A. Ferrari and S. Yuhjtman.

For a string of length n, the overlapping function defines the greatest size of a repetition, in the sense that it is k if its first and last k symbols coincide. When the source that generates the strings satisfies the complete grammar condition, the overlapping function is always non-negative.

In the present work we deal with the case where the complete grammar condition is removed, and therefore “negative overlaps” are allowed. We state a weak convergence theorem when the source is a beta-mixing Markov Chain with finite diameter (greatest “distance” between two symbols of the alphabet). This is a work in progress in collaboration with Erika Alejandra Rada-Mora (UFABC).

We study the stochastic system of interacting neurons introduced in De Masi et al 2015, in a diffusive scaling. The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to 0 and all other neurons receive an additional amount of potential which is a centred random variable of order 1 / sqrt (N). In between successive spikes, each neuron’s potential follows a deterministic flow. We prove the convergence of the system, as the number of neurons tends to infinity, to a limit nonlinear jumping stochastic differential equation driven by Poisson random measure and an additional Brownian motion W which is created by the central limit theorem. This Brownian motion is underlying each particle’s motion and induces a common noise factor for all neurons in the limit system.

Conditionally on W, the different neurons are independent in the limit system. This is the “conditional propagation of chaos” property. We prove the well-posedness of the limit equation by adapting the ideas of Graham 1992 to our frame. To prove the convergence in distribution of the finite system to the limit system, we introduce a new martingale problem that is well suited for our framework. The uniqueness of the limit is deduced from the exchangeability of the underlying system.

This is a joint work with Xavier Erny and Dasha Loukianova, both of university of Evry.

In this talk we investigate a one-dimensional infinite mechanical particle system, driven by a constant force F. The system consists of one charged particle, together with field-neutral ones. Neutral particles are initially randomly placed in the medium, and can be perfectly elastic or inelastic, according to independent Bernoulli random variables. We establish central limit theorems for the velocity and position of the charged particle.

Based on joint work with Luiz Renato Fontes and Rémy Sanchis.

We study d-dimensional random walks in strong mixing environments (RWRE), with underlying dimension d>=2. Under a suitable polynomial effective condition, we prove a functional central limit theorem of ballistic type. Specifically, we construct a new effective criterion equivalent to usual ballisticity conditions. This construction allows us to prove, in a mixing framework, the RWRE conjecture regarding the equivalence between ballisticity conditions already proved for iid environments. We then obtain the polynomial effective condition that provides the existence of arbitrary finite moments for approximate regeneration times, yielding the central limit theorem for the RWRE.

Joint work with Maria Eulalia Vares (UFRJ) and Enrique Guerra (PUC-Chile)

A well known result implies that the rescaled maximal height of a system of N non-intersecting Brownian bridges starting and ending at the origin converges, as N goes to infinity, to the Tracy-Widom GOE random variable from random matrix theory. In this talk I will focus on the same question in case where the top m paths start and end at arbitrary locations. I will present several related results about the distribution of the limiting maximal height for this system, which provides a deformation of the Tracy-Widom GOE distribution: it can be expressed through a Fredholm determinant formula and in terms of Painlevé transcendents; it corresponds to the asymptotic fluctuations of models in the KPZ universality class with a particular initial condition; and it is connected with two PDEs, the KdV equation and an equation derived by Bloemendal and Virag for spiked random matrices. Based on joint work with Karl Liechty and Gia Bao Nguyen

Garsia–Rodemich–Rumsey (1971) proved an inequality which has been used to upper-bound the Brownian modules of continuity. In turn, this upper-bound was used by T.G. Kurtz (1976) to prove a strong diffusion approximation result for pure jump processes. However, this proof makes the crucial assumption that jump rates are uniformly bounded.
The main objective of this talk is to show how to get rid of this assumption starting back from GRR inequality. The second objective is to show that this scheme of proof is robust to time scaling.

The Contact Process was introduced by Harris in 1974 and models the spread of an infection on a graph. The state of each vertex is either infected or healthy, and there are two competing factors that govern the evolution of the process over time: infected vertices become healthy at rate 1 and healthy vertices can get infected at a rate proportional to its current number of infected neighbors. In two recent papers, Fontes, Marchetti, Mountford and Vares introduced a generalization of the model in which cures are given by renewal processes with some fixed interarrival distribution. I will discuss how the choice of interarrival distribution affects the spread of the infection, focusing on recent developments in which we improved the characterization of the interarrival distributions for which there is phase transition. Joint work with Luiz Renato Fontes, Tom Mountford and Maria Eulália Vares.