Seminário de Probabilidade e Análise Estocástica 

06/12/2023, 16:00 — 17:00 — Online
Alessia Nota, University of L'Aquila

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)

29/11/2023, 16:00 — 17:00 — Online
Alberto Chiarini, University of Padova

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).

15/11/2023, 16:00 — 17:00 — Online
Sonia Velasco, Université Paris-Cité

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).

08/11/2023, 16:00 — 17:00 — Online
Arjun Krishnan, University of Rochester, New York

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)

25/10/2023, 17:00 — 18:00 — Online
Avelio Sepúlveda, Universidad de Chile

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.

18/10/2023, 17:00 — 18:00 — Online
Paul Chleboun, University of Warwick

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).

11/10/2023, 17:00 — 18:00 — Online
Michael Conroy, University of Arizona

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.

04/10/2023, 17:00 — 18:00 — Online
Davide Gabrielli, Università degli Studi dell'Aquila

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.

27/09/2023, 17:00 — 18:00 — Online
Adrián González Casanova, University of California at Berkeley

Sample Duality

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.

20/09/2023, 17:00 — 18:00 — Online
Matteo D'Achille, LMO Université Paris-Saclay

Almost Gibbsian Measures on a Cayley Tree

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)

05/07/2023, 17:00 — 18:00 — Online
Alessandra Occelli, Université d'Angers

Universality of multi-component stochastic systems

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.

28/06/2023, 17:00 — 18:00 — Online
Márton Balázs, University of Bristol

Blocking measures is a combinatorial goldmine

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)

21/06/2023, 17:00 — 18:00 — Online
Assaf Shapira, Université Paris Cité

Topologically induced metastability in periodic XY chain

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.

14/06/2023, 17:00 — 18:00 — Online
Partha Dey, University of Illinois, Urbana-Champaign

Curie-Weiss Model under $l^p$ constraint

We consider the Curie-Weiss model on the complete graph $K_n$ with spin configurations constrained to have a given $l^p$ norm for some $p>0$. For $p=\infty$, this reduces to the classical Ising Curie-Weiss model. We generalize the model with a self-scaled Hamiltonian for general symmetric spin distribution with variance one. Using a modified Hubbard-Stratonovich transform and a coupling of log-gamma distributions, we compute the limiting free energy. As a consequence, we prove that for all $p>1$, there exists a critical $\gb_c(p)$ such that for $\gb<\gb_c(p)$, the magnetization is concentrated at zero and satisfies a Gaussian CLT. In contrast, the magnetization is not concentrated at zero for $\gb>\gb_c(p)$, similar to the classical case. While $\gb_c(2)=1$, we have $\gb_{c}(p)>1$ for $p>2$. To understand the magnetization, we introduce an exchangeable dynamics on the $l^p$ sphere surface, which is of independent interest. For $0 < p < 1$, the log-partition function scales at the order of $n^{2/p-1}$. Based on joint work with Daesung Kim.

07/06/2023, 17:00 — 18:00 — Online
Clément Erignoux, INRIA - Lille

Modelling active matter by active lattice gases: exact hydrodynamic description and phase transitions

In this talk, I will introduce a few related microscopic models for active matter. The models we consider are lattice gases, meaning that the active particles jump stochastically on a lattice. Their active nature is represented by a drift in their stochastic jumps, whose direction can evolve in time as particles interact with eachother. I will discuss how, with this type of lattice gases, one can model the behavior of active matter, and recover the emergence of Vicsek's alignment phase transition as well as Motility Induced Phase Separation (MIPS), both classical phenomena for active matter. Both have been well documented by the physics community, however mathematical results remain scarce. Notably, using the mathematical theory of hydrodynamic limit, one can prove the emergence of both phenomena mathematically, even for models with purely local interactions, without any mean-field type assumptions. I will talk about recent results on phase separation occuring in a non gradient active gas, and how even small proportion of active particles can induce phase separation. Weak solutions to the homogeneous Boltzmann equation with increasing energy have been constructed by Lu and Wennberg. We consider an underlying microscopic stochastic model with binary collisions and show that these solutions are atypical. More precisely, we prove that the probability of observing these paths is exponentially small in the number of particles and compute the exponential rate. Based on JW with Mourtaza Kourbane Houssène, Julien Tailleur, Thierry Bodineau, James Mason, Maria Bruna, Robert Jack.

31/05/2023, 17:00 — 18:00 — Online
Sylvie Méléard, École Polytechnique

24/05/2023, 17:00 — 18:00 — Online
Lorenzo Bertini, La Sapienza - Roma

On the probability of observing energy increasing solutions to the Boltzmann equation

Weak solutions to the homogeneous Boltzmann equation with increasing energy have been constructed by Lu and Wennberg. We consider an underlying microscopic stochastic model with binary collisions and show that these solutions are atypical. More precisely, we prove that the probability of observing these paths is exponentially small in the number of particles and compute the exponential rate. Weak solutions to the homogeneous Boltzmann equation with increasing energy have been constructed by Lu and Wennberg. We consider an underlying microscopic stochastic model with binary collisions and show that these solutions are atypical. More precisely, we prove that the probability of observing these paths is exponentially small in the number of particles and compute the exponential rate. Weak solutions to the homogeneous Boltzmann equation with increasing energy have been constructed by Lu and Wennberg. We consider an underlying microscopic stochastic model with binary collisions and show that these solutions are atypical. More precisely, we prove that the probability of observing these paths is exponentially small in the number of particles and compute the exponential rate. Weak solutions to the homogeneous Boltzmann equation with increasing energy have been constructed by Lu and Wennberg. We consider an underlying microscopic stochastic model with binary collisions and show that these solutions are atypical. More precisely, we prove that the probability of observing these paths is exponentially small in the number of particles and compute the exponential rate.

17/05/2023, 17:00 — 18:00 — Online
Alex Dunlap, New York University

The nonlinear stochastic heat equation in the critical dimension

I will discuss a two-dimensional stochastic heat equation with a nonlinear noise strength, and consider a limit in which the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor. The limiting pointwise statistics can be related to a stochastic differential equation in which the diffusivity solves a PDE somewhat reminiscent of the porous medium equation. This relationship is established through the theory of forward-backward SDEs. I will also explain several cases in which the PDE can be solved explicitly, some of which correspond to known probabilistic models. This talk will be based on current joint work with Cole Graham and earlier joint work with Yu Gu.

10/05/2023, 17:00 — 18:00 — Online
Kevin Yang, UC Berkeley

Universality and well-posedness for a time-inhomogeneous KPZ equation

The KPZ equation is a model for non-equilibrium interface fluctuations that comes from perturbing a Gaussian Langevin dynamic by a slope-dependent nonlinearity. An associated universality problem is whether or not the same model comes from (scaling limits of) perturbing non-Gaussian "Ginzburg-Landau" SDEs by a slope-dependent nonlinearity. One goal of this talk is to give a "fluctuation version" of Yau's relative entropy method to resolve this problem for a general class of non-Gaussian potentials. The microscopic models considered also have a non-equilibrium flavor that leads to a time-inhomogeneous KPZ equation, which introduces its own interesting mathematics at both the microscopic and macroscopic levels.

03/05/2023, 17:00 — 18:00 — Online
Federico Sau, University of Trieste

Spectral gap of the symmetric inclusion process

In this talk, we consider the symmetric inclusion process on a general finite graph. Our main result establishes universal upper and lower bounds for the spectral gap of this interacting particle system in terms of the spectral gap of the random walk on the same graph. In the regime in which the gamma-like reversible measures of the particle system are log-concave, our bounds match, yielding a version for the symmetric inclusion process of the celebrated Aldous' spectral gap conjecture - originally formulated for the interchange process and proved by Caputo, Liggett and Richthammer (JAMS 2010). Finally, by means of duality techniques, we draw analogous conclusions for an interacting diffusion-like unbounded conservative spin system known as Brownian energy process, which may be interpreted as a spatial version of the Wright-Fisher diffusion with mutation. Based on a joint work with Seonwoo Kim (SNU, South Korea).