Probability and Stochastic Analysis Seminar 

Stein's method is an increasingly popular way to derive quantitative versions of weak convergence theorems, like the central limit theorem. In this talk we use Stein's method to derive a quantitative CLT for Dynkin martingales of Markov chains. Despite its simplicity, we show with some examples that the bounds we obtain in the context of interacting particle systems are surprisingly sharp.

I will talk about a singular non-linear SPDE that was introduced by van Beijeren, Kutner and Spohn (1985) as a continuum version of d-dimensional ASEP. The equation is "supercritical" ($d>3$) or critical ($d=2$) in the SPDE language. We show that the large-scale behavior of the equation is Gaussian in dimension $d$ greater or equal to $3$ (this mirrors analogous results by Landim, Olla, Yau et al for ASEP) and also in dimension $d=2$ (in the so-called weak noise limit, which corresponds to a certain $2-$dimensional WASEP). The scaling is non-trivial in the sense that the non-linearity has a non-vanishing effect on the limit equation. Ongoing work with G. Cannizzaro, L. Haunschmid and M. Gubinelli.

In this talk, first, I give a very brief introduction to the NLS (Nonlinear Schrödinger Equation) and its long-time behavior. Then I introduce a mass-conserving stochastic perturbation of the discrete nonlinear Schrödinger equation that models the action of a heat bath at a given temperature. Afterward, I sketch the fact that the corresponding canonical Gibbs distribution is the unique invariant measure. Finally, as an application, I discuss the one-dimensional cubic focusing case on the torus, where we prove that in the limit for large time, continuous approximation, and low temperature, the solution converges to the steady wave of the continuous equation that minimizes the energy for a given mass. This is based on the following joint work with prof. Stefano Olla (Universite Paris Dauphine-PSL, GSSI, IUF):
Hannani, A., Olla, S. A stochastic thermalization of the Discrete Nonlinear Schrödinger Equation. Stoch PDE: Anal Comp (2022). https://doi.org/10.1007/s40072-022-00263-9

This talk focuses on generalizations of the exclusion process whose hydrodynamic limits are sub-diffusive equations. After recalling some known results in dimension 1, I will present in detail the partial exclusion process in random environment. This is a system of random walks where the random environment is obtained by assigning random maximal occupancies to each site of the Euclidean lattice. I will show that, when assuming that the maximal occupancies are heavy tailed and i.i.d., the hydrodynamic limit of the particle system (in any dimension greater than 1) is the fractional-kinetics equation.

This talk is based on partly ongoing projects in collaboration with A. Chiarini (Padova), F. Redig (TU Delft) and F. Sau (ISTA).

In this talk we will consider two models within Kardar-Parisi-Zhang (KPZ) universality class, and apply the integration by parts formula from Malliavin calculus to establish a key relation between the two-point correlation function, the polymer end-point distribution and the second derivative of the variance of the associated height function. Besides that, we will further develop an adaptation of Malliavin-Stein method that quantifies asymptotic independence with respect to the initial data.

We consider a system of N half lines issuing from the origin, on which there is a Poisson process of dust particles initially. We have N^ \alpha workers who clean dust particles according to a greedy algorithm; they move to the closest dust particle and remove it and then wait an exponential time before chosing a new particle. We consider for which values of alpha one can have half lines where two or more workers go to infinity. The talk uses only elementary probability arguments and well known properties of Poisson processes. It should be accessible to all.
Joint with Sergey Foss and Takis Konstantopoulous

The Axelrod model is a spatial stochastic model for the dynamics of cultures which includes two important social components: homophily, the tendency of individuals to interact more frequently with individuals who are more similar, and social influence, the tendency of individuals to become more similar when they interact. Each individual is characterized by a collection of opinions about different issues, and pairs of neighbors interact at a rate equal to the number of issues for which they agree, which results in the interacting pair agreeing on one more issue. This model has been extensively studied during the past 20 years based on numerical simulations and heuristic arguments while there is a lack of analytical results. This talk gives rigorous fluctuation and fixation results for the one-dimensional system that sometimes confirm and sometimes refute some of the conjectures formulated by applied scientists.

In this joint work with M. Falconnet and N. Gantert, we define interacting particle systems on configurations of the integer lattice (with values in some finite alphabet) by the superimposition of two dynamics: a substitution process with finite range rates, and a circular permutation mechanism (called “cut-and-paste”) with possibly unbounded range.

The model is motivated by the dynamics of DNA sequences: we consider an ergodic model for substitutions, the RN+YpR model, introduced by Berard et al. in 2008, as well as three particular cases. We investigate whether they remain ergodic with the additional cut-and-paste mechanism, which models insertions and deletions of nucleotides. Using either duality or attractiveness techniques, we provide various sets of sufficient conditions, concerning only the substitution rates, for ergodicity of the superimposed process.

The exclusion process is one of the best-studied examples of an interacting particle system. In this talk, we consider simple exclusion processes on finite graphs. We give an overview over some recent results on the mixing time of the totally asymmetric simple exclusion process (TASEP). In particular, we provide bounds on the mixing time of the TASEP on the circle, using a connection to periodic last passage percolation. This talk is based on joint work with Allan Sly.

Understanding the diffusive or superdiffusive behavior of the energy in classical physical systems is challenging because of the non-linearity of the interactions between the particles. A way to reduce the difficulty is to replace the nonlinearity by a stochastic noise. In this presentation I will consider a noisy harmonic chain subjected to a magnetic field. We will see that according to the intensity of the magnetic field, the superdiffusive nature of the system changes.

The Kardar-Parisi-Zhang (KPZ) equation is a stochastic partial differential equation with universality, and it has been derived from several microscopic models through scaling limits. When the temperature of a system tends to infinity, we can often extract a heat diffusion part with some residual perturbation by a Taylor expansion argument, which decomposition is crucial for the derivation. We will show through some particular models that we can thereby obtain the KPZ equation as a limit in a robust way.

Consider a sequence of continuous-time Markov chains $X^n_t$ evolving on a fixed finite state space $V$. Let $I_n$ be its level two large deviations rate functiona. Under a general hypothesis on the jump rates, we derive an expansion for $I_n$: we show that $I_n$ can be written as$ I^0 + \sum_{1\ le p\le q} (\theta^p_n)^{-1} I^p$ In this formula, $\theta^p_n$ are the time-scales at which a meta-stable behavior is observed and I^p the level two large deviations rate function of the Markov chain which describes the evolution of the chain $X^n_t$ in the time-scale $\theta^p_n$.

We will study a model of noisy units in mean field interaction, which is described in the large population limit by a non linear PDE. Relying on a slow/fast approach we will show the emergence of noise induced periodic behaviors. We will focus in particular on the case when each unit evolves according to the FiztHugh Nagumo model.

Spin glasses are disordered spin systems originally introduced to explain some unusual magnetic behavior of certain alloys. Although their formulations are typically easy to state, many of them enjoy several profound mathematical and physical principles that are extremely challenging to prove. In this talk, I will focus on the Sherrington-Kirkpatrick model and consider its local magnetization in the computational perspective. While it has been well-understood that this quantity satisfies so-called cavity method and Thouless-Anders-Palmer equations in the literature, I will explain how they give rise to novel iterative algorithms and are helpful in simulating the local magnetization in the high temperature regime. Based on a joint work with Si Tang.

The Kardar-Parisi-Zhang (KPZ) equation is the stochastic partial differential equation that models interface growth. In the talk I will present the construction of a stationary measure for the KPZ equation on a bounded interval with general inhomogeneous Neumann boundary conditions. Along the way, we will encounter classical orthogonal polynomials, the asymmetric simple exclusion process, and precise asymptotics of q-Gamma functions. This construction is a joint work with Ivan Corwin.

We will introduce a periodic version of the Polynuclear growth model (PNG) and show that it is a solvable model. We can give stationary measures for the model at a fixed time and for the distribution of the space-time paths, which in this model are up-down paths that form rings. This is joint work with Pablo Ferrari (UBA).

We discuss some recent developments in the analysis of convergence to stationarity for the Gibbs sampler of general spin systems on arbitrary graphs. These are based on two recently introduced concepts: Spectral Independence and Block Factorization of Entropy. We show that the existence of a contractive coupling for a local Markov chain implies that the system is spectrally independent, and that if a system is spectrally independent then its entropy functional satisfies a general block factorization. As a corollary, we obtain new optimal bounds on the mixing time of a large class of sampling algorithms for the ferromagnetic Ising/Potts models in the so-called tree-uniqueness regime, including non-local chains such as the Swendsen-Wang dynamics. The methods apply to systems with hard constraints such as proper colorings and the hard core gas. We also discuss the entropy factorization for the uniform distribution over permutations and its role in the proof of a conjectured bound on the permanent of arbitrary matrices. Based on some recent joint works with Alexandre Bristiel, Antonio Blanca, Zongchen Chen, Daniel Parisi, Alistair Sinclair, Daniel Stefankovic, and Eric Vigoda.

I will review results on fluctuations of the height function (particle density) for the open Asymmetric Simple Exclusion Process in steady state as the size of the system goes to infinity. I will discuss the cases where the parameters of ASEP are constant or vary with the size of the system. The talk is based on several papers with Alexey Kuznetsov, Yizao Wang and Jacek Wesolowski.

This talk aims to report on the fluctuations of traces of powers of a random $n$ by $n$ matrix U distributed according to the Haar measure on the unitary group. This random matrix problem has been extensively studied using several different methods such as asymptotics of Toeplitz determinants, representation theory, loop equations etc. It turns out that for any $k≥1$, $Tr[U^k]$ converges as $n$ tends to infinity to a Gaussian random variable with a super exponential rate of convergence. In this talk, I will explain some of these results and present some recent work with Klara Courteaut and Kurt Johansson (KTH) in which we revisited this classical problem.

The cylinder’s percolation model arises from a Poissonian soup of infinite lines in $R^d$ and it is a stationary process under the isometries of the underlying space. Each such line is then thickened, becoming the axis of a cylinder of radius one. The associated percolation picture exhibits long range correlations and the rigidity of the underlying objects hampers direct attempts at proving decorrelation inequalities via sprinkling of the intensity parameter. We obtain such inequalities by exploiting the continuity of the process, taking the radii of the cylinders as a parameter and using it in a sprinkling argument. As an application, we prove that for small intensities of the cylinder’s process the simple random walk on the vacant set is transient. The talk will also go over similar decoupling inequalities for other models, their applications and open problems.
This talk is based on a joint work with Caio Alves.