Probability and Stochastic Analysis Seminar 

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.

Smoluchowski’s coagulation equation, an integro-differential equation of kinetic type, is a classical mean-field model for mass aggregation phenomena. The solutions of the equation exhibit rich behavior depending on the rate of coagulation considered, such as gelation (formation of particles with infinite mass in finite time) or self-similarity (preservation of the shape over time). In this talk I will first discuss some fundamental properties of the Smoluchowski equation. I will then present some recent results on the problem of existence or non-existence of stationary solutions, both for single and multi-component systems, under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. The most striking feature of these stationary solutions is that, whenever they exist, the solutions to multi-component systems exhibit an unusual “spontaneous localization” phenomenon: they localize along a line in the composition space as the total size of the particles increase. This localization is a universal property of multicomponent systems and it has also been recently proved to occur in time dependent solutions to mass conserving coagulation equations. 
(Based on joint works with M. Ferreira, J. Lukkarinen and J. Velázquez)

In this talk we aim at establishing large deviation estimates for the probability that a simple random walk on the Euclidean lattice ($d\gt 2$) covers a substantial fraction of a macroscopic body. It turns out that, when such rare event happens, the random walk is locally well approximated by random interlacements with a specific intensity, which can be used as a pivotal tool to obtain precise exponential rates. Random interlacements have been introduced by Sznitman in 2007 in order to describe the local picture left by the trace of a random walk on a large discrete torus when it runs up to times proportional to the volume of the torus, and has been since a popular object of study. In the first part of the talk we introduce random interlacements and give a brief account of some results surrounding this object. In the second part of the talk we study the event that random interlacements cover a substantial fraction of a macroscopic body. This allows to obtain an upper bound on the probability of the corresponding event for the random walk. Finally, by constructing a near-optimal strategy for the random walk to cover a macroscopic body, we discuss a matching large deviation lower bound. The talk is based on ongoing work with M. Nitzschner (NYU Courant).

We derive a formula for the quasi-potential of the one-dimensional symmetric exclusion process in weak contact with reservoirs. The interaction with the boundary is so weak that, in the diffusive scale, the density profile evolves as the one of the exclusion process with reflecting boundary conditions. In order to observe an evolution of the total mass, the process has to be observed in a longer time-scale, in which the density profile becomes immediately constant. This is joint work with Claudio Landim (IMPA).

In dimension 1, the directed polymer model is in the celebrated KPZ universality class, and for all positive temperatures, a typical polymer path shows non-Brownian KPZ scaling behavior. In dimensions 3 or larger, it is a classical fact that the polymer has two phases: Brownian behavior at high temperature, and non-Brownian behavior at low temperature. We consider the response of the polymer to an external field or tilt, and show that at fixed temperature, the polymer has Brownian behavior for some fields and non-Brownian behavior for others. In other words, the external field can *induce* the phase transition in the directed polymer model. (joint work with S. Mkrtchyan and S. Neville)

Joint work with Christophe Garban. The discrete Coulomb gas is a model where an integer amount of charged particles are put on the d-dimensional grid. In this talk, I will discuss the fundamental properties of the Coulomb gas through its connection with other statistical physics models. In particular, I will focus on its ergodic properties, its scaling limit and the so-called Debye screening.

We consider facilitated exclusion processes (FEP) in one dimension. These models belong to a class of kinetically constrained lattice gases. The process was introduced in the physics literature motivated by studying the active-absorbing phase transition. Under the dynamics, a particle can move to a neighbouring site provided that the target site is empty (the exclusion rule) and the other neighbour of the departure site is occupied (the constraint). These processes have recently attracted a lot of attention due to their interesting hydrodynamic limit behaviour. We examine the mixing time, the time to reach equilibrium, on an interval with closed boundaries and also with periodic boundary conditions. On the interval we observe that asymmetry significantly changes the mixing behaviour. The analysis naturally splits into examining the time to reach the ergodic configurations (irreducible component) followed by the time needed to mix on this set of configurations. This is joint work with James Ayre (Oxford).

In the one-dimensional exclusion system, a step initial condition is one with infinitely many particles to the left and none to the right of a maximal one. Assuming symmetric, nearest-neighbor interaction, if we tag the right-most particle and follow its (properly scaled) position as time grows, we see a Gumbel limit distribution. Interestingly, this matches the behavior of the maximum of independent particles started from the same initial profile, as studied by Arratia (1983). Unlike with independent particles, proving the result for the exclusion process requires a careful analysis of pair-wise correlations, which rests on duality and negative association properties of symmetric exclusion. Limiting Gumbel distributions can also be obtained in higher dimensions by considering initial conditions where infinitely many particles occupy points in a half-space. This talk is based on joint work with Sunder Sethuraman.

Boundary driven stochastic lattice gases are simple but effective models for non equilibrium statistical mechanics. Apart special cases, as for example the zero range model where the stationary state is always of product type, they exhibit long range correlations. I will discuss a class of models for which it is possible, in the boundary driven case, to give a simple representation of the invariant measure in terms of mixtures of inhomogeneous product measures. This is true for the Kipnis Marchioro Presutti model and its dual and for a class of generalized zero range dynamics.

Heuristically, two processes are dual if one can find a function to study one process by using the other. Sampling duality is a duality which uses a duality function S(n,x) of the form "what is the probability that all the members of a sample of size n are of a certain type, given that the number (or frequency) of that type of individuals is x". Implicitly, this technique can be traced back to the work of Blaise Pascal. Explicitly, it was studied in a paper of Martin Möhle in 1999 in the context of population genetics. We will discuss examples for which this technique is useful, including an application to the Simple Exclusion Process with reservoirs.

The ferromagnetic Ising model on infinite regular trees has a longstanding tradition in Probability and Statistical Mechanics. As such, it offers a solid benchmark in the quest for putting Renormalization Group ideas from Physics on rigorous grounds. In this talk, I will introduce a mapping on Ising configurations on the 3-regular infinite tree, namely a modified majority rule transformation, which was already known to lead to non-Gibbsian measures at any temperature. However, we show that the renormalized measure, whose properties can be studied thanks to a model of percolation of zeros, actually satisfies at any temperature an almost sure version of Gibbsianity, which we call almost-Gibbsianity. Key ingredients of the discussion will be the celebrated Kozlov-Sullivan Theorem for Gibbsian specifications, the recursivity inherent to the treatment on trees and temperature-dependent bond percolation. Talk mostly based on a joint paper with Arnaud Le Ny (Markov Process. Relat. Fields 28, 2022)

Universality classes are identified by exponents and scaling functions that characterise the macroscopic behaviour of the fluctuations of the thermodynamical quantities of interest in a microscopic system. When considering multi-component systems different universality classes might appear according to the asymmetry of the interactions. To see which universality classes might appear, we outline the approach of Nonlinear Fluctuation Hydrodynamics Theory (NLFHT), introduced by Spohn 2014. As an example, we study the equilibrium fluctuations of an exclusion process evolving on the discrete ring with three species of particles named $A$, $B$ and $C$. We prove that proper choices of density fluctuation fields (that match of those from nonlinear fluctuating hydrodynamics theory) associated to the conserved quantities converge, in the large $N$ limit, to a system of stochastic partial differential equations, that can either be the Ornstein-Uhlenbeck equation or the Stochastic Burgers' equation.

Several asymmetric nearest-neighbour interacting particle systems possess reversible product stationary distributions called blocking measures. Whatever we ask about these a new proof of a non-trivial combinatorial identity drops out as a result. Simple exclusion's particle-hole symmetry, the number of its particles in parts of the integer line, or the exclusion-zero range correspondence each give rise to probabilistic proofs of partition identities (namely, Durfee Rectangles Identity, Euler's Identity, the q-Binomial Theorem, Jacobi Triple Product). More complicated systems beyond simple exclusion can also be studied, and these provide more involved combinatorial results, some of them completely new. I'll reveal some structures behind blocking measures, and sketch how to prove a bunch of scary-looking identities using interacting particles, hence bringing them closer to a probabilist. As a by-product the stationary location of simple exclusion's second class particles in blocking measures will also be revealed. (Joint with Dan Adams, Ross Bowen, Dan Fretwell, Jessica Jay)

Many physical phenomena are explained using statistical physics models with non-trivial topological properties. One of the most important models showing this type of behavior is the XY model, which in two dimensions possesses a topological phase transition. The model discussed in this talk is the simpler one-dimensional XY model, in a low temperature regime where topological observables could be identified. We consider the dynamics of this model, explain over which time scales these observables change, and identify a temperature regime in which the equilibrium has no topological order, but the dynamics allows for metastable states with non-trivial topology. Based on a joint work with Clément Cosco.