N. Gozlan, O. Guédon, A. Guionnet, C. Léonard, C. Roberto, F. Redig, B. Zegarlinski

We prove that Gaussian dimension free concentration is equivalent to Talagrand's T2 transportation-cost inequality. The proof of this result relies on the Large Deviations approach developped with C. Léonard. A nice corollary of this result is a short and elementary new proof of Otto and Villani's Theorem stating that the Logarithmic-Sobolev inequality is stronger than T2. Other results can be proved in the same spirit. For example, we shall see that Poincaré inequality is equivalent to a certain type of dimension free exponential concentration property.

Olivier Guédon (Paris): Independance versus convexity. The singular values of some random matrices.

The goal of this talk will be to present results in probability indicating that the independance of the coordinates of a random vector can be replaced by the fact that the random vector is uniformly distributed on a convex body in high dimension. In particular, we will present new results concerning the smallest singular value of a random matrix defined with i.i.d. columns drawn from a vector uniformly distributed on a symmetric convex body.

Alice Guionnet (Lyon): Gibbs measures on U(N) and O(N); concentration of measures and law of large numbers.

We shall consider Gibbs measures which are absolutely continuous with respect to the Haar measure on O(N) or U(N) and with a small potential. We shall obtain self-averaging of normalized traces of polynomial in these random matrices, almost sure convergence and give a combinatorial interpretation of the limit.

Christian Léonard (Nanterre): Transport inequalities and large deviations.

We give a survey of some results connecting transport inequalities with large deviations of sequences of independent and identically distributed random variables and Markov processes. This connection allows to extend several transport inequalities in different directions. It also provides a probabilistic interpretation of these inequalities. The contributors to this recent approach are N. Gozlan, A. Guillin, A. Joulin, C. Léonard and L. Wu.

Cyril Roberto (Marne-La-Vallée): Isoperimetry for product probability measures.

In this talk we shall give a short overview on the isoperimetric problem for product probability measures.

An isoperimetric inequality is a lower bound on the boundary measure of sets in terms of their measure. Finding the optimal sets (of given measure and of minimal boundary measure) is very difficult, and the only hope is to estimate the isoperimetric function. This is well understood on the line (Bobkov) and for the product of standard Gaussian measures (Sudakov-Tsirel'son, Borell). We shall start by recalling those known results.

Then, we shall explain how functional inequalities can be used to get dimension free isoperimetric inequalities for measures between exponential and Gaussian. Also, using the transport of mass technique we shall dervive isoperimetric inequalities (depending on the dimension) for measures with tails larger than exponential.

Frank Redig (Eindhoven): Concentration via coupling.

We present a method based on martingale difference approach combined with coupling to obtain concentration inequalities for Lipschitz functions in the context of models of statistical mechanics (Gibbs measures). Estimates are in terms of a coupling matrix. The regime where the coupling matrix can be estimated uniformly corresponds to Gaussian inequalities, in the non-uniform regime weaker inequalities such as moment inequalities can be obtained. This is illustrated for the low-temperature Ising model. The talk is based on joint work with J.R. Chazottes, P. Collet, and C. Kuelske.

Boguslaw Zegarlinski (Toulouse): Linear and Nonlinear Concentration Phenomena.

I plan to describe some problems and results concerning certain linear and nonlinear effects in large interacting systems.