Recent advances on log-gases

Institut Henri Poincaré, Paris - Friday March 21, 2014 - Amphi Hermite

Five 45' talks for a wide audience on recent advances on log-gases.
The talks will be given in amphitheatre Hermite (venue: IHP).

A Coulomb gas is simply a system of interacting particles in which the interaction is of Coulomb type (repulsive). Replacing the Coulomb repulsion by a more general (singular) interaction leads to the versatile concept of log-gas. These stochastic models appear naturally in classical physics and analysis. They are at the heart of several high dimensional universality phenomena related to the spectral analysis of random matrix models. They also appear in other fields such as complex geometry and partial differential equations. To learn more on these models, you may come! The registration is free. Here is the program:

• 09:30-10:00 Welcome
• 10:00-10:45 Arno Kuijlaars (Leuven) Normal matrix model and Laplacian growth. PDF
  Eigenvalues in the normal matrix model can be considered as a 2d log-gas, that in the large n limit fill out a domain in the complex plane. The boundary of the domain evolves according to the model of Laplacian growth. In interesting cases, however, the model is not well-defined. I will explain a way to regularize the model (joint work with P. Bleher) and to obtain detailed information in a supercritical regime (joint work with A. Tovbis).
• 10:45-11:15 Break
• 11:15-12:00 Robert Berman (Göteborg) A large deviation principle for non linear log-gas. PDF
  In this talk I will explain the proof of a Large Deviation Principle (LDP) for the large N-limit of the Gibbs measures associated to rather general singular N-particle Hamiltonian. In a nutshell the result says that if an LDP holds in the zero-temperature regime, then there is also an LDP for any fixed positive temperature, where the new rate functional is obtained by adding an entropy contribution to the zero-temperature rate functional. This is an essentially well-known result of mean field type in the “linear case”, i.e. when the N-particle Hamiltonian is a sum of two-points functions. However, the proof in the general case involves some rather sharp geometric analysis and Riemannian geometry on the corresponding N-particle spaces. As will be briefly pointed out the LDP in question is motivated by a probabilistic approach to the construction of Kähler-Einstein metrics on complex algebraic varieties, which involves higher dimensional generalizations of the normal random matrix model (see arXiv:1307.3634)
• 12:00-14:00 Break
• 14:00-14:45 Sylvia Serfaty (Paris) Energy approach to Coulomb and log gases. PDF
  I will describe works in which we perform a next order expansion of the Hamiltonian for Coulomb gases in all dimensions and log gases in dimension 1. At that next order we are able to derive a limiting “renormalized” energy, corresponding to the total Coulomb energy of an infinite “jellium”, computed on the microscopic patterns formed by the points. This energy is conjectured to be minimized by certain crystalline configurations. Equidistribution of points and energy can be proven for ground states of the Hamiltonian. For the situation with temperature, we can deduce a next order expansion of the partition function and crystallization-type results. This is based on joint works with Etienne Sandier, Nicolas Rougerie and Simona Rota Nodari.
• 14:45-15:00 Break
• 15:00-15:45 Paul Bourgade (Princeton) Microscopic structure of log-gases in dimension 1. (blackboard talk)
  I will explain works with L. Erdős and H.-T. Yau on (1) local Gibbs measures for 1d log-gases, and on (2) the application to universality for generalized Wigner matrices. An important tool is the rigidity of log-gases obtained up to the edge, thanks to a local log-Sobolev inequality.
• 15:45-16:15 Break
• 16:15-17:00 Gaëtan Borot (Bonn) Large-N asymptotic expansions in 1-d repulsive particle systems. PDF
  By 1-d repulsive particle systems, I mean systems of N particles on the real line, repelling each other with 2d Coulomb repulsion, and subjected to an analytic k-body interaction. I will review general techniques based on Schwinger-Dyson equation and concentration of measures to obtain the asymptotic expansion when N → infinity of moments and partition function, and address the question of convergence in law of fluctuations of smooth linear statistics. One has to distinguish the one-cut regime (expansion in 1/N) and the multi-cut regime (oscillatory expansion). I will also describe the current limits of this approach.The talk is based on joint works with A. Guionnet and K. Kozlowski.arXiv:1312.6664

Registration is over.

You might by interested by Limite de champ moyen et condensation de Bose-Einstein by Mathieu Lewin (Gazette des Mathématiciens 139, 35-49, Société Mathématique de France), and by the Cours Peccot in Collège de France of Nicolas Rougerie on the same subject.

• Djalil Chafaï (Paris-Dauphine)
• Nathaël Gozlan (Paris-Est)
• Pierre-André Zitt (Paris-Est)
  

• Institut Henri Poincaré (IHP)
• Institut Universitaire de France (IUF)