Topics on Log and Coulomb Gases

We are interested in systems of points with Coulomb, logarithmic or more generally Riesz interactions (i.e. inverse powers of the distance). They arise in the study of some random matrix ensembles and so-called beta-ensembles. We will take a point of view based on the detailed expansion of the interaction energy to describe the microscopic behavior of the systems. In particular a Central Limit Theorem for fluctuations and a Large Deviations Principle for the microscopic point processes will be described. This allows to observe the effect of the temperature as it gets very large or very small, and to connect with crystallization questions. Based on joint works with Etienne Sandier, Nicolas Rougerie, Mircea Petrache, Thomas Leblé, Florent Bekerman and Scott Armstrong. on the statistical mechanics of systems of points with logarithmicor Coulomb interactions. After listing some motivations, we describe the “electric approach"which allows to get concentration results, Central Limit Theorems for fluctuations, and aLarge Deviations Principle expressed in terms of the microscopic state of the system.

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