Seminário de Probabilidade e Análise Estocástica 

Bootstrap percolation is a classical statistical physics model displaying metastable behaviour. Let each site of the square lattice be infected independently with a fixed probability. At each round, infect each site with at least two infected neighbours and do not remove any infections. How long does it take before the origin is infected? We start by reviewing the rich history of this problem and some of the classical arguments used to tackle it. We then give a very precise answer to the above question in the relevant regime of sparse infection. The key to the proof is a new locality approach to bootstrap percolation, which also resolves the bootstrap percolation paradox concerning the failure of numerical predictions in the field. The talk is based on joint work with Augusto Teixeira available at https://arxiv.org/abs/2404.07903.

We consider a large class of supercritical spatially embedded random graphs, including among others long-range percolation and geometric inhomogeneous random graphs, and identify a single exponent zeta depending on the model parameters that describes the asymptotics of

1. the probability that the largest connected component is much smaller than expected;
2. the size of the second-largest component;
3. the distribution of the size of the component containing a distinguished vertex.

In the talk, I will explain the relation between the three quantities and give some intuition for the values of zeta in different regimes.

Joint work with Joost Jorritsma and Júlia Komjáthy.

The Directed Polymer in a Random Environment is a statistical mechanics model, which has been introduced (in dimension 1) as a toy model to study the interfaces of the planar Ising model with random coupling constants. The model was shortly afterwards generalized to higher dimensions. In this latter case, rather than an effective interface model, the directed polymer in a random environment can be thought of as modeling the behavior of a stretched polymer in a solution with impurities. The interest in the model model, triggered by its rich phenomenology, has since then generated a plentiful literature in theoretical physics and mathematics. An important topic for the directed polymer is the so-called localization transition. This transition can be defined in terms of the asymptotic behavior of the renormalized partition function of the model. If the finite volume partition function converges to an almost surely positive limit we say that weak disorder holds. On the other hand, if it converges to zero almost surely, we say that strong disorder holds. It has been proved that weak disorder implies that the distribution of the rescaled polymer converges to standard Brownian motion while some localization results have been proved under the strong disorder assumptions. Much stronger characterizations of disorder-induced localization have been obtained under the stronger assumption that the partition function converges to zero.

(joint with Stefan Junk, Gakushuin University)

We consider large systems of jump processes that interact locally with respect to an underlying (possibly random) graph. Such processes model diverse phenomena including the spread of diseases, opinion dynamics and gas dynamics. Under a broad set of assumptions, we show that the empirical measure satisfies a large deviation principle in the sparse regime, that is, when the sequence of graphs converges locally to a limit graph. As a corollary we establish (quenched) hydrodynamic limits for the sequence of interacting jump processes. In addition, for a sub-class of processes that include the SIR process, we obtain a fairly explicit characterization of this limit and provide numerical evidence to show that it serves as a good approximation for finite systems of moderate size.

This is based on various joint works with I-Hsun Chen, Juniper Cocomello and Sarath Yasodharan.

Thanks to the theories of Regularity Structures, Paracontrolled Distributions and Renormalisation we now have a robust framework for singular SPDEs, which are “sub-critical” in the sense of renormalisation. Recently, there have been efforts to approach the situation of “critical” SPDEs and statistical mechanics models. A first such treatment has been through the study of the two-dimensional stochastic heat equation, which has revealed a certain phase transition and has led to the construction of the novel object called the Critical 2d Stochastic Heat Flow. In this talk we will present some aspects of this model and its construction. We will also present developments relating to other critical SPDEs.
Parts of this talk are based on joint works with Caravenna and Sun and others with Rosati and Gabriel.

We derive the heat equation for the thermal energy under diffusive space-time scaling from a purely deterministic microscopic dynamics satisfying Newton equations perturbed by an external chaotic force acting like a magnetic field.

Joint work with Giovanni Canestrari and Carlangelo Liverani.

We study the $\Phi_2^3$-measure in the infinite volume limit. This Gibbs-type measure inspired by similar objects appearing in QFT also correspondis to the invariant measure for the nonlinear SPDE known as the stochastic quantization equations. We give sharp estimates for the partition function in the large torus limit which imply strong concentration of the field around zero.

Joint work with K. Seong.

Physicists and mathematicians have for decades been interested in the Abelian sandpile model (ASM), a simple dynamical system that exhibits self-organized criticality. In the present work we asked us the question: what function of a Gaussian field produces the same scaling limit of its joint moments as the height-one field of the ASM? The squared norm of the gradient of the discrete Gaussian free field (GFF) almost does the job, with a subtle yet important difference. It turns out that, if one replaces the variables by Grassmann (or fermionic) objects, we obtain the exact same moments, which begs the follow-up question: why? What is the relationship between these two models? We will show that this relationship lies on the uniform spanning tree (UST) model. This observation will allow us to extend the calculations to square lattices in any dimensions, as well as the triangular and hexagonal lattices in d=2, pointing towards a potential universality property. We also extend our results by giving exact closed-form expressions for the scaling limit of the PMF of the degree field of the UST in these lattices. Joint work with L. Chiarini (U Durham), A. Cipriani (UCL), R.S. Hazra (U Leiden), W.M. Ruszel (U Utrecht).

A cellular automaton describes a process in which cells evolve according to a set of rules. Which rule is applied to a specific cell only depends on the states of the neighboring and the cell itself. Considering a one-dimensional cellular automaton with finite range, the positive rates conjecture states that and under the presence of noise the associated stationary measure must be unique. We restrict ourselves to the case of nearest-neighbor interaction where simulations suggest that the positive rates conjecture is true. After discussing a simple criterion to deduce decay of correlations, we show that the positive rates conjecture is true for almost all nearest-neighbor cellular automatons. The main tool is comparing a one-dimensional cellular automaton to a properly chosen two-dimensional Ising-model. We outline a pathway to resolve the remaining open cases. This presentation is based on collaborative work with Maciej Gluchowski from the University of Warsaw and Jacob Manaker from UCLA

In this talk, we will consider two models for diffusing particles in time-dependent random environments: the discrete random walk in random environment (RWRE) and a continuum scaling limit of the RWRE called sticky Brownian motion. We will present some recent results on the weak convergence of both models to the KPZ equation in the moderate deviation regime. We will also discuss an application to the fluctuations of the maximal particle in these models. This is joint work with Sayan Das and Shalin Parekh.

Birkhoff's ergodic theorem is a cornerstone in Mathematics, with a huge range of applications. We present here an extended form of the multidimensional ergodic theorem with weights, which allows to derive large scale averaging of stationary ergodic random measures on $\mathbb{R}^d$, also when the testing observables are not compactly supported. This control at infinity of random measures plays a crucial role when analysing the large scale behaviour of RWs and IPSs on weighted random graphs on $\mathbb{R}^d$ built on simple point processes. In particular, this allows us to obtain homogenization and hydrodynamics under weaker conditions.

We will then discuss several examples in order to emphasize the universality of our results.

The model of directed polymers describe the behavior of a long, directed chain that spreads among an inhomogeneous environment which may attract or repulse the polymer. When the spacial dimension is larger than three, a phase transition occurs between diffusivity (high temperature) and localization (low temperature). On the other hand, in dimensions one and two the polymer is always localized. Dimension two is however critical, as one can recover a phase transition by letting the temperature tend to infinity under a specific parametrization (Caravenna-Sun-Zygouras 17’). In this talk, I will present some of the main results that are known about this scaling regime, and discuss the recent advances that have occurred in the past few years. In particular, I will describe some results that I have obtained with my coauthors (Anna Donadini, Shuta Nakajima and Ofer Zeitouni) on the diffusive phase and its relation to Gaussian logarithmically correlated fields. I will also discuss connexions of the model with the Kardar-Parisi-Zhang (KPZ) equation and the stochastic heat equation.

We introduce a novel variant of the exclusion process where particles make asymmetric nearest neighbor jumps across a bond (k,k+1) only when the site k-1 to the left of the bond is empty. This next-nearest-neighbor interaction significantly enriches the model's behavior. We show that for a system with periodic boundary conditions, ergodicity is ensured only for systems that are strictly less than half-filled. For half-filling the system segregates into two distinct ergodic components, and we provide the invariant measure for each component and prove that it is reversible. The combinatorial properties of this invariant measure are intimately related to the q-Catalan numbers, where q represents the asymmetry of the two elementary hopping events. Exploiting this relation allows us to extract the asymptotic behavior both in the strongly and weakly asymmetric regimes. We report a phase separation characterized by different critical exponents for which we provide an intuitive geometrical interpretation. This is joint work with Gunter Schütz.

The goal of my talk is to present some results on the Ising model on a Galton-Watson tree. I will start with a general introduction, recalling in particular the seminal results of Russell Lyons. I will the present the results obtained in collaboration with Irene Ayuso Ventura (University Paris-Est Créteil), which estimate the effect of a sparse external field or boundary condition on the magnetisation of the root.

We consider a random walk jumping on a dynamic graph; that is, a graph that changes at the same time as the walker moves. Previous works considered the case where the graph changes via dynamical percolation, in which the edges of the graph switch between two states, open and closed, and the walker is only allowed to cross open edges. In dynamical percolation, edges change their state independently of one another. In this work, we consider a graph dynamics with unbounded dependences: Glauber dynamics on the random cluster model. We derive tight bounds on the mixing time when the density of open edges is small enough. For the proof, we construct a non-Markovian coupling using a multiscale analysis of the environment. This is based on joint work with Andrea Lelli.

A cellular automaton describes a process in which cells evolve according to a set of rules. Which rule is applied to a specific cell only depends on the states of the neighboring and the cell itself. Considering a one-dimensional cellular automaton with finite range, the positive rates conjecture states that and under the presence of noise the associated stationary measure must be unique. We restrict ourselves to the case of nearest-neighbor interaction where simulations suggest that the positive rates conjecture is true. After discussing a simple criterion to deduce decay of correlations, we show that the positive rates conjecture is true for almost all nearest-neighbor cellular automatons. The main tool is comparing a one-dimensional cellular automaton to a properly chosen two-dimensional Ising-model. We outline a pathway to resolve the remaining open cases. This presentation is based on collaborative work with Maciej Gluchowski from the University of Warsaw and Jacob Manaker from UCLA

To study fluctuations in the weakly asymmetric simple exclusion process at large space scale $x\varepsilon^{-1}$, large time scale $t \varepsilon^{-\chi}$ and weak hopping bias $b \varepsilon^{\kappa}$ in the limit $\varepsilon \to 0$ we develop a mesoscale mode coupling theory (MMCT) that allows for probing the crossover at $\kappa=1/2$ and $\chi=2$ from Kardar-Parisi-Zhang (KPZ) to Edwards-Wilkinson (EW) universality. The dynamical structure function is shown to satisfy an integral equation that is independent of the microscopic model parameters and has a solution that yields a scale-invariant function with the KPZ dynamical exponent $z=3/2$ at scale $\chi=3/2+\kappa$ for $0\leq\kappa<1/2$ and for $\chi=2$ the exact Gaussian EW solution with $z=2$ for $\kappa>1/2$. At the crossover point it is a function of both scaling variables which converges at macroscopic scale to the conventional mode coupling approximation of KPZ universality for $\kappa<1/2$. This fluctuation pattern confirms long-standing conjectures for $\kappa \leq 1/2$ and is in agreement with mathematically rigorous results for $\kappa>1/2$ despite the numerous uncontrolled approximations on which mode coupling theory is based.

We survey several aspects of random matrices, and how they connect to different branches of mathematics. In particular, we plan to explain an unexpected connection we recently found with the so-called Kardar-Parisi-Zhang (KPZ) equation.

The "Discrete Gaussian Chain" is a model which extends the celebrated long-range 1D lsing model with $1/r^\alpha$ interactions. The latter model is known to have a rather intriguing phase-diagram. Instead of having +/- spins, the discrete Gaussian Chain is a random field with values in the integers Z. After introducing this model and its history, I will describe its large scale fluctuations and will compare its phase diagram with the case of long-range Ising model.

The aim of this talk is to give a heuristic understanding of an intriguing extension of the classical TASEP. Instead of only considering nearest-neighbour jumps, we allow for arbitrary jumps with a rate that decays polynomially in the distance. The talk concentrates on new results from joint work with P. Gonçalves and L. Xu for the boundary-driven version of this model.