From Boltzmann to random matrices and beyond

Chafaï, Djalil

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Numéro Spécial : Conférence “Talking Across Fields” du 24 au 28 mars 2014 à l’Institut de Mathématiques de Toulouse, Tome 24 (2015) no. 4, pp. 641-689.

Ces notes d’exposition proposent de suivre, à travers différents domaines, quelques aspects du concept d’entropie. À partir du travail de Boltzmann en théorie cinétique des gas, plusieurs univers sont visités, incluant les processus de Markov et leur énergie libre de Helmholtz, le problème de Shannon de monotonie de l’entropie dans le théorème central limite, la théorie des probabilités libres de Voiculescu et le théorème central limite libre, les marches aléatoires sur les arbres réguliers, la loi du cercle pour l’ensemble de Ginibre complexe de matrices aléatoires, et enfin l’analyse asymptotique de systèmes de particules champ moyen en dimension arbitraire, confinées par un champ extérieur et subissant une répulsion singulière à deux corps. Le texte est écrit dans un style informel piloté par l’énergie et l’entropie. Il vise a être récréatif, à fournir aux lecteurs curieux des points d’entrée dans la littérature, et des connexions au delà des frontières.

These expository notes propose to follow, across fields, some aspects of the concept of entropy. Starting from the work of Boltzmann in the kinetic theory of gases, various universes are visited, including Markov processes and their Helmholtz free energy, the Shannon monotonicity problem in the central limit theorem, the Voiculescu free probability theory and the free central limit theorem, random walks on regular trees, the circular law for the complex Ginibre ensemble of random matrices, and finally the asymptotic analysis of mean-field particle systems in arbitrary dimension, confined by an external field and experiencing singular pair repulsion. The text is written in an informal style driven by energy and entropy. It aims to be recreative and to provide to the curious readers entry points in the literature, and connections across boundaries.

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