Probability and Stochastic Analysis Seminar 

In the study of scaling limits of reversible particle systems with the property of self-duality, many quantities of interest become easier to manipulate and simplify. For the particular case of fluctuations from the hydrodynamic limit, and in the additional presence of orthogonality, these simplifications have interesting consequences. In this talk, we will briefly discuss some of those consequences. First, we will obtain a systematic orthogonal decomposition of the fluctuation fields of local functions, where the order of every term can be quantified. This implies a quantitative generalization of the Boltzmann–Gibbs principle. Second we will introduce what we call the k-th order fluctuation field. We will then explain how these fields can be interpreted as some type of discrete analogue of powers of the well-known density fluctuation field, and show how their scaling limits formally correspond to the SPDE associated with the kth-power of a generalized Ornstein-Uhlenbeck process.

This work takes inspiration from [1] and [2], and it is a joint effort with G. Carinci (Università di Modena e R. Emilia) and Frank Redig (TU Delft).

1. Sigurd Assing, A limit theorem for quadratic fluctuations in symmetric simple exclusion, Stochastic Process. Appl. 117 (2007), no. 6, 766–790.
2. Patrícia Gonçalves and Milton Jara, Quadratic fluctuations of the symmetric simple exclusion, ALEA Lat. Am. J. Probab. Math. Stat. 16 (2019), no. 1, 605–632.

Random intersection graph is a simple random graph that incorporates community structures. To build such a graph, imagine there are $n$ individuals and $m$ potential communities. Each individual joins a community independently with probability $p$. The graph $G(n, m, p)$ has $n$ nodes, corresponding to the $n$ individuals. Each pair of these individuals share an edge between them if they belong to a common community. The critical threshold for the emergence of a giant component emerges turns out to be at $p^2 ~ 1/nm$. I’ll discuss some results that can help us to understand what a large $G(n, m, p)$ looks like at the critical threshold.

The canonical framework for mean-field systems is to consider $N$ particles (diffusions, or dynamics with jumps, etc) that interact on the complete graph in a uniform way, the strength of interaction between two particles being of size $1/N$. The behavior of the system is hence captured by the empirical measure of the system which converges as $N\to\infty$ to the solution of a nonlinear Fokker Planck equation. The motivation of this talk is simple: what can we say if one no longer interacts on the complete graph, i.e. one removes connections between particles ? if the graph is sufficiently close to the complete graph, one expects the same asymptotic behavior. We will address this question at the level of the LLN and fluctuations of the empirical measure of the system. This is based on joint works with G. Giacomin, S. Delattre, F. Coppini and C. Poquet.

Generalized Ray-Knight theorems for edge local times proved to be a very useful tool for studying the limiting behavior of several classes of self-interacting random walks (SIRWs) on integers. Examples include some reinforced random walks, excited random walks, rotor walks with defects. I shall describe two classes of SIRWs introduced and studied by Balint Toth (1996), asymptotically free and polynomially self-repelling SIRWs, and discuss new results which resolve an open question posed in Toth’s paper. We show that in the asymptotically free case the rescaled SIRWs converge to a perturbed Brownian motion (conjectured by Toth) while in the polynomially self-repelling case the convergence to the conjectured process fails in spite of the fact that generalized Ray-Knight theorems clearly identify the unique candidate in the class of all perturbed Brownian motions. This negative result was somewhat unexpected. The question whether there is convergence in the polynomially self-repelling case and, if yes, then how to describe the limiting process is open. This is joint work with Thomas Mountford, EPFL, and Jonathon Peterson, Purdue University.

In this talk we will review a model that was first introduced by Jonasson, Mossel and Peres. Starting with the usual square lattice on $Z^2$, entire rows (respectively columns) of edges extending along the horizontal (respectively vertical) direction are removed independently at random. On the remaining thinned lattice, Bernoulli bond percolation is performed, giving rise to a percolation model with infinite range dependencies under the annealed law. In 2005, Hoffman solved the main conjecture around this model: proving that this percolation process indeed undergoes a nontrivial phase transition. In this talk, besides reviewing this surprisingly challenging problem, we will present a novel proof, which replaces the dynamic renormalization presented previously by a static version. This makes the proof easier to follow and to extend to other models. We finally present some remarks on the sharpness of Hoffman’s result as well as a list of interesting open problems that we believe can provide a renewed interest in this family of questions. This talk is based on a joint work with M. Hilário, M. Sá and R. Sanchis.

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.