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From Boltzmann to random matrices and beyond

Persi Diaconis during his talk, Toulouse, March 2014
Persi during his talk – Toulouse, March 2014.

I have recently uploaded on the French HAL a preprint numbered hal-00987177 (on arXiv it is 1405.1003) modestly entitled From Boltzmann to random matrices and beyond. This is my first upload in the History and Overview (math.HO) section. This initial version has 26 pages. Remarkably, the number $26$, used for Applied Maths by the French Conseil national des universités, is the unique number between a square ($25=5^2$) and a cube ($27=3^3$), cf. [GB].

This text forms the written notes of a talk entitled “About confined particles with singular pair repulsion”, given at the occasion of the workshop “Talking Across Fields” on convergence to equilibrium of Markov chains. This exciting workshop was organized in Toulouse from 24 to 28 March by Laurent Miclo at the occasion of the CIMI Excellence research chair for Persi Diaconis.

Regarding the content, 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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