From Boltzmann to random matrices and beyond
Chafaï, Djalil (2015), From Boltzmann to random matrices and beyond, Annales de la Faculté des Sciences de Toulouse, 24, 4, p. 641-689. 10.5802/afst.1459
TypeArticle accepté pour publication ou publié
External document linkhttps://arxiv.org/abs/1405.1003v4
Journal nameAnnales de la Faculté des Sciences de Toulouse
Université Paul Sabatier
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Abstract (EN)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.
Subjects / KeywordsPotential theory; Random matrices; Circular law; Boltzmann; Electrostatics; Collective phenomena; Mean-field interaction; Coulomb gas; Voiculescu; Ginibre ensemble; Interacting Particle Systems; Entropy; Large Deviations Principle; Riesz kernel; Shannon; Equilibrium measure; Singular repulsion
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