Séminaire Images Optimisation et Probabilités

Mercredi 10/07

14h00 Jurgen Angst (Univ. Rennes)

Title :  TLC in total variation for beta-ensembles

Abstract : In this talk, we study the fluctuations of linear statistics associated with beta-ensembles, which are statistical physics models generalizing random matrix spectra. In the context of random matrices precisely (e.g. GOE, GUE), the "law of large numbers" is Wigner's theorem, which states that the empirical measure of eigenvalues converges to the semicircle law, and fluctuations around equilibrium can be shown to be Gaussian. We will describe how this result generalizes to beta-ensembles and how it is possible to quantify the speed of convergence to the normal distribution. We obtain optimal rates of convergence for the total variation distance and the Wasserstein distances. To do this, we introduce a variant of Stein's method for a generator $L$ that is not necessarily invertible, and which allows us to establish the asymptotic normality of observables that are not in the image of $L$. Time permitting, we will also look at the phenomenon of super-convergence, which ensures that convergence to the normal law takes place for very strong metrics, typically the $C^{\infty}$-convergence of densities. The talk is based on recent works with R. Herry, D. Malicet and G. Poly.

15h00 Nicolas Juillet (Univ. Haute-Alsace)

Title :  Exact interpolation of 1-marginals

Abstract : I shall present a new type of martingales that exactly interpolates any given family of 1-dimensional marginals on R1 (satisfying the suitable necessary assumption). The construction makes use of ideas from the (martingale) optimal transportation theory and relies on different stochastic orders. I shall discuss of related constructions and open questions (joint work with Brückerhoff and Huesmann).

16h00 Kolehe Coulibaly-Pasquier (Inst. Ellie Cartan)

Title :  On the separation cut-off phenomenon for Brownian motions on high dimensional rotationally

symmetric compact manifolds.

Abstract : Given a family of compact, rotationally symmetric manifolds indexed by the dimension and a weighted function, we will study the cut-off phenomena for the Brownian motion on this family.

Our proof is based on the construction of an intertwined process, a strong stationary time, an estimation of the moments of the covering time of the dual process, and on the phenomena of concentration of the measure.

We will see a phase transition concerning the existence or not of cut-off phenomena, which depend on the shape of the weighted function.

Jeudi 11/07

11h00 Masha Gordina (Univ. of Connecticut)

Title :   Dimension-independent functional inequalities on sub-Riemannian manifolds

Abstract : The talk will review recent results on gradient estimates, log Sobolev inequalities, reverse Poincare and log Sobolev inequalities on a class of sub-Riemannian manifolds. As for many of such setting curvature bounds are not available, we use different techniques including tensorization and taking quotients. Joint work with F. Baudoin, L. Luo and R. Sarkar.

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