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10/04/2024, 17:00 — 18:00 — Online

Georg Menz, *University California Los Angeles*

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The Positive Rates Conjecture in Nearest-Neighbor Cellular Automata

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

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03/04/2024, 17:00 — 18:00 — Online

Hindy Drillick, *Columbia University*

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The KPZ equation limit of random walks in random environments

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.

Please note the different time in Europe.

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27/03/2024, 16:00 — 17:00 — Online

Alessandra Faggionato, *University La Sapienza*

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An ergodic theorem with weights and applications to random measures, RW homogenisation and IPS hydrodynamics.

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.

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20/03/2024, 16:00 — 17:00 — Online

Clement Cosco, *Université Paris Dauphine*

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Directed polymers in random environment and the critical dimension

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.

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06/03/2024, 16:00 — 17:00 — Online

Ali Zahra, *IST-Lisboa*

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Asymmetric exclusion process with next nearest neighbor interaction

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.

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28/02/2024, 16:00 — 17:00 — Online

Quentin Berger, *LPSM, Sorbonne Université, and DMA, Ecole Normale Supérieure, Paris*

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Some results about the Ising model on a Galton-Watson tree

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.

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21/02/2024, 16:00 — 17:00 — Online

Alexandre Stauffer, *King's College London*

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Mixing time of random walk on dynamical random cluster

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.

Except for a few of the oldest sessions these are from the Seminário de Probabilidade e Mecânica Estatística at IMPA which is co-sponsored by several institutions, in particular Instituto Superior Técnico.

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14/02/2024, 16:00 — 17:00 — Online

Georg Menz, *University of California Los Angeles*

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The Positive Rates Conjecture in Nearest-Neighbor Cellular Automata

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

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07/02/2024, 16:00 — 17:00 — Online

Gunter Schütz, *Forschungszentrum Jülich*

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Mesoscale mode coupling theory for the weakly asymmetric simple exclusion process

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.

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24/01/2024, 16:00 — 17:00 — Online

Guilherme Silva, *ICMC, USP São Carlos*

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Random matrices: from Painlevé to KPZ

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.

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17/01/2024, 16:00 — 17:00 — Online

Christophe Garban, *Camille Jordan Institute*

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Fluctuations of the Discrete Gaussian Chain

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.

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20/12/2023, 16:00 — 17:00 — Online

Julian Kern, *Weierstrass Institute*

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TASEP's big brother: the totally asymmetric long jump asymmetric exclusion process (TALJEP)

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.

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06/12/2023, 16:00 — 17:00 — Online

Alessia Nota, *University of L'Aquila*

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On the Smoluchowski equation for aggregation phenomena: stationary non-equilibrium solutions

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)

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29/11/2023, 16:00 — 17:00 — Online

Alberto Chiarini, *University of Padova*

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How efficiently does a simple random walk cover a portion of a macroscopic body?

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).

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15/11/2023, 16:00 — 17:00 — Online

Sonia Velasco, *Université Paris-Cité*

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Quasi-potential for the one dimensional SSEP in weak contact with reservoirs

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).

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08/11/2023, 16:00 — 17:00 — Online

Arjun Krishnan, *University of Rochester, New York*

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On the phase diagram of the polymer model

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)

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25/10/2023, 17:00 — 18:00 — Online

Avelio Sepúlveda, *Universidad de Chile*

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On the discrete Coulomb gas

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.

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18/10/2023, 17:00 — 18:00 — Online

Paul Chleboun, *University of Warwick*

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Mixing times for Facilitated Exclusion Processes

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).

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11/10/2023, 17:00 — 18:00 — Online

Michael Conroy, *University of Arizona*

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Extreme values in the symmetric exclusion process

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.

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04/10/2023, 17:00 — 18:00 — Online

Davide Gabrielli, *Università degli Studi dell'Aquila*

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Solvable stationary non equilibrium states

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.