Probability and Stochastic Analysis Seminar 

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).

26/04/2023, 17:00 — 18:00 — Online
Fraydoun Rezakhanlou, UC - Berkeley

Kinetic Theory for Laguerre Tessellations

In this talk I will discuss a family of Gibbsian measures on the set of Laguerre tessellations. These measures may be used to provide a systematic approach for constructing Gibbsian solutions to Hamilton-Jacobi PDEs by exploring the Eularian description of the shock dynamics. Such solutions depend on kernels satisfying kinetic-like equations reminiscent of the Smoluchowski model for coagulating and fragmenting particles.

19/04/2023, 17:00 — 18:00 — Online
Giuseppe Cannizzaro, University of Warwick and UKRI Future Leaders Fellow

Crossover from the Brownian Castle to Edwards-Wilkinson

In the context of randomly fluctuating interfaces in (1+1)-dimensions two Universality Classes have generally been considered, the Kardar-Parisi-Zhang (KPZ) and the Edwards-Wilkinson (EW). Notoriously, the KPZ equation is known to interpolate between them in that its small-scale statistics are those of EW while its large-scale fluctuations are those of KPZ. In a recent work with M. Hairer, we showed that the universality picture outlined above is not exhaustive and identified a new universality class together with the universal process at its core, the Brownian Castle (BC). After reviewing the origin, construction and characterising properties of BC, the talk will be devoted to show that there exist a huge family of processes that play a role similar to that of the KPZ equation, connecting though the BC and EW universality classes. We called these processes $\nu$-Brownian Castle, for $\nu$ a probability measure on [0,1], and are linked to the Brownian Net and the stochastic flows of kernels of Schertzer, Sun and Swart. Time allowing, we will show that (one of) these processes naturally arise as the limit, under a suitable scaling, of a microscopic model given by a stochastic PDE. This is joint (ongoing) work with M. Hairer, T. Rosati and R. Sun.

12/04/2023, 17:00 — 18:00 — Online
Timo Seppalainen, University of Wisconsin

Stationary horizon as the universal multitype stationary distribution

The stationary horizon (SH) is a recently constructed cadlag stochastic process whose states are Brownian motions and the process is indexed by the drifts. It is part of the universality picture of the 1+1 dimensional Kardar-Parisi-Zhang (KPZ) class. SH was discovered as a diffusive limit of the Busemann process of the exponential corner growth model (Busani) and simultaneously as the Busemann process of Brownian last-passage percolation (Sorensen and the speaker). This talk is about SH as the Busemann process of the directed landscape, as the stationary distribution of the KPZ fixed point, and as the scaling limit of the TASEP speed process. Joint work with Ofer Busani (Bonn/Edinburgh) and Evan Sorensen (Madison/Columbia).

05/04/2023, 17:00 — 18:00 — Online
Chiara Franceschini, University of Modena

Two duality relations for Markov processes with an open boundary

In this talk I will show how the same algebraic approach, which relies on the $\mathfrak{su}(1,1)$ Lie algebra, can be used to construct two duality results. One is well-known: the two processes involved are the symmetric inclusion process and a Markov diffusion called Brownian Energy process. The other one is a new result which involves a particle system of zero-range type, called harmonic process, and a redistribution model similar to the Kipnis-Marchioro-Presutti model. Despite the similarity, it turns out that the second relation involves integrable models and thus duality can be pushed further. As a consequence, all moments in the stationary nonequilibrium state can be explicitly computed.

29/03/2023, 16:00 — 17:00 — Online
Stefan Grosskinsky, Augsburg University

Size-biased diffusion limits for the inclusion process 

We study the Inclusion Process with vanishing diffusion coefficient, which is known to exhibit condensation and metastable dynamics for cluster locations. Here we focus on the dynamics of mass distribution rather than locations, and consider the process on the complete graph in the thermodynamic limit with fixed particle density. We describe the mass distribution for a given configuration by a measure on a suitably scaled mass space and derive a limiting measure-valued process. When the diffusion coefficient scales like the inverse of the system size, the scaling limit is equivalent to the well known Poisson-Dirichlet diffusion, offering an alternative point of view for this well-established dynamics. Testing configurations with size-biased functions, our approach can be generalized to other scaling regimes. This leads to an interesting characterization of the limiting dynamics via duality and provides a natural extension of the Poisson-Dirichlet diffusion to infinite mutation rate. This is joint work with Simon Gabriel and Paul Chleboun (both Warwick)

22/03/2023, 16:00 — 17:00 — Online
Benoit Dagallier, Cambridge University

Large deviations for out of equilibrium correlations in the symmetric simple exclusion process 

For finite size Markov chains, the probability that a time-averaged observable take an anomalous value in the long time limit was quantified in a celebrated result by Donsker and Varadhan. In the study of interacting particle systems, one is interested not only in the large time, but also in the large system size limit. In this second limit, observables of the chain each live at different scales, and one has to understand how scales decouple. In a joint work with Thierry Bodineau, we study a paradigmatic example of out of equilirium interacting particle systems: the one-dimensional symmetric simple exclusion process connected with reservoirs of particles at different density. We focus on the scale of two point correlations and obtain the long time, large system size limits on the probability of observing anomalous correlations. This is done through quantitative, non-asymptotic estimates at the level of the dynamics. The key ingredient is a precise approximation of the dynamics and its invariant measure (not explicitly known), that is of independent interest. The quality of this approximation is controlled through relative entropy bounds, making use of recent results of Jara and Menezes.

15/03/2023, 16:00 — 17:00 — Online
Linjie Zhao, Wuhan University

The tagged particle in asymmetric exclusion process with long jumps

In this talk, we consider the asymmetric exclusion process with long jumps, where the transition rate is proportional to $ |x|^{-d-\alpha} $ for some $ \alpha > 0 $. We state the central limit theorems for the tagged particle when the process starts from its stationary measure. When time permitted, we will also outline the proof.

08/03/2023, 16:00 — 17:00 — Online
Jinho Baik, University of Michigan

Multi-point distribution of periodic TASEP and differential equations

We discuss random growth models in the KPZ universality on a ring. When the time and the size of the ring both tend to infinity in a critical way, the height fluctuation field is expected to converge to a field that interpolate the KPZ fixed point on the line and the Brownian motion. We discuss the convergence of the multi-time, multi-position distributions for the totally asymmetric simple exclusion process. In the second part of the talk, we discuss various deterministic differential equations associated with the KPZ fixed point and its periodic version for the narrow-wedge initial condition.