# Exposés

Les rencontres seront composées de deux mini-cours, de conférences de 45 minutes, ainsi que de sessions posters.

## Mini-cours

• Massimiliano GUBINELLI (CEREMADE, Université Paris Dauphine).
Singular Stochastic PDEs and paracontrolled distributions.

• Bálint TÓTH (School of Mathematics, University of Bristol, England; Institute of Mathematics, TU, Budapest, Hungary).
Scaling limits for random walks and diffusions with long memory - resolvent methods.

## Posters

• Kevin KUOCH (Université Paris Descartes).
• Achref MAJID (Faculté des sciences de Tunis El Manar).
• Arnaud ROUSSELLE (Université de Rouen).

## Liste des participants

À télécharger ici.

## Résumés

• Mohamed-Amine ASSELAH (LAMA, Université Paris-Est).
On two deposition models.

I discuss properties of a two models of stochastic deposition: ballistic and diffusive limited deposition.
• Christophe BAHADORAN, (Université Blaise Pascal, Clermont Ferrand).
Phase transition for disordered TASEP.

We prove that the current-density relation of TASEP with site disorder exhibits a plateau around density 1/2 for sufficiently dilute disorder. This result was first conjectured by Tripathy and Barma (1998). The difficulty is that invariant measures are unknown in the case of site disorder contrary to particle disorder, where the existence of a linear portion on the current can be established by explicit computation. Our proof uses a renormalization scheme and last passage percolation. This is a joint work with T. Bodineau.
• Cédric BERNARDIN(Université de Nice Sophia-Antipolis).
3/4 fractional superdiffusion of energy in a harmonic chain with bulk noises.

We consider a harmonic chain perturbed by an energy conserving noise and show that after a space-time rescaling the energy-energy correlation function is given by the solution of a skew-fractional heat equation with exponent 3/4.
• Thierry BODINEAU (CNRS, CMAP, École Polytechnique).
Tagged particle in a deterministic dynamics of hard spheres.

We consider a tagged particle in a diluted gas of hard spheres. Starting from the hamiltonian dynamics of particles in the Boltzmann-Grad limit, we will show that the tagged particle follows a Brownian motion after an appropriate rescaling. We use the linear Boltzmann equation as an intermediate level of description for one tagged particle in a gas close to global equilibrium.
• Ivan CORWIN (Clay Mathematics Institute, Columbia University, Institute Henri Poincare, Massachusetts Institute of Technology, Cambridge, USA).
Macdonald processes, quantum integrable systems and the KPZ universality class.

A large class of one dimensional systems are predicted to share the same universal long-time/large-scale behaviors. By studying certain integrable models within this Kardar-Parisi-Zhang (KPZ) universality class we access what should be universal statistics and phenomena. The purpose of this talk is to give an introductory overview of how representation theory and integrable systems can be harnessed in the form of the theory of Macdonald processes and quantum integrable systems to discover and analyze a variety of probabilistic models (such as directed polymers, interacting particle systems, growth processes, random matrices, and tilings).
• Alexandre GAUDILLIERE (CNRS, LATP, Université d'Aix Marseille)
Pyramidal algorithms for general graphs or metastability without asymptotic.

We propose some adaptation for general graphs of usual pyramidal algorithms of signal processing to bridge between standard techniques of multiscale analysis and usual approaches to metastability. This makes possible to deal wtih metastable dynamics outside any asymptotic regime. This is work in progress with Luca Avena, Fabienne Castel and Clothilde Melot.
• Massimiliano GUBINELLI (CEREMADE, Université Paris Dauphine).
Singular Stochastic PDEs and paracontrolled distributions.

These two lectures will be devoted to the introduction of an analytic framework for the formulation of non-linear evolution problems perturbed by singular noise sources which can arise as scaling limits of certain microscopic evolutions. The parabolic anderson model, the Kardar-Parisi-Zhang equation and the stochastic quantization equation are examples of such systems. Solving (or even giving a meaning to) these equations require a detailed understanding of the propagation of the stochastic perturbations via the non-linear evolution. I will explain how ideas and tools from harmonic analysis can be useful in this analysis and in the related problem of studying the convergence of the microscopic models to their scaling limits.
• Régine MARCHAND (Institut Elie Cartan, Université de Lorraine, Nancy).
Boolean percolation in high dimension.

In the boolean percolation model in $\mathbb{R}^d$, random balls are thrown in the following manner:
- centers are chosen accordingly to a homogeneous Poisson point process with intensity $\lambda$ in $\mathbb{R}^d$,
- radii are iid random variables with common distribution $\nu$, also independent from the centers.
We say that percolation occurs if there exists an infinite connected component of balls: there is a critical parameter $\lambda_c(\nu)$ such that if $\lambda <\lambda_c(\nu)$, there is a.s. no percolation, while if $\lambda>\lambda_c(\nu)$, percolation occurs with positive probability.
We investigate here the distribution of radius $\nu$ which minimizes the critical parameter $\lambda_c(\nu)$.
In particular, by studying the asymptotic of this critical parameter $\lambda_c(\nu)$ when $\nu$ only charges two distinct radii, we disprove the conjecture that constant radii should be optimal.
• Stefano OLLA (CEREMADE, Université Paris Dauphine).
A Wigner distribution approach to hydrodynamic limits: diffusion and superdiffusion of energy.

• Mathew PENROSE (University of Bath, England).
Random parking and rubber elasticity.

Renyi's random parking process on a domain D in d-space is a point process with hard-core and no-empty-space properties that are desirable for modelling materials such as rubber. It is obtained as follows: particles arrive sequentially at uniform random locations in D, and are rejected if they violate the hard-core constraint, until the accepted particles saturate D.
We describe how any real-valued functional on this point process, provided it enjoys certain subadditivity properties, satisfies an averaging property in the thermodynamic limit. Consequently in this limit, one has a convergence of macoroscopically-defined energy functionals for deformations of the point process, to a homogenized limiting energy functional. We may also apply the results to derive laws of large numbers for classical optimization problems such as travelling salesman on the parking point process. This is joint work with Antoine Gloria.
• Christophe SABOT (Institut Camille Jordan, Université Claude Bernard, Lyon).
The Vertex Reinforced Jump Process and the inversion of the generalized Ray-Knight identity.

The vertex reinforced jump process (VRJP) is a self-interacting process closely related to the Edge Re- inforced Random Walk. The generalized Ray-Knight identity relates the local time of a Markov jump process to the associated Gaussian free field. After a review of recent results on these processes I will explain how the Ray-Knight identity can be "inverted" using ideas coming from the VRJP.
• Renato Soares Dos SANTOS (Institut Camille Jordan, Université Claude Bernard, Lyon).
Random walk on random walks.

We study a random walk in a dynamical random environment given by a Poissonian system of independent random walks in equilibrium, for which we prove a law of large numbers and a central limit theorem in a large density regime.
• Bálint TÓTH (School of Mathematics, University of Bristol, England; Institute of Mathematics, TU, Budapest, Hungary).
Scaling limits for random walks and diffusions with long memory - resolvent methods.

I will survey recent results on diffusivity bounds and central limit theorem for random walks and diffusions with path-wise self-interaction and for random walks and diffusions in a particular class of random environment. The results rely on analysis of the infinitesimal generator and resolvent of the semigroup of Markov processes related to the walks/diffusions in question. The lectures will be self-contained, I will survey classical and new results related to additive functionals of ergodic Markov processes.