Lectures on Coulomb and Riesz Gases

This volume provides a comprehensive introduction to the mathematical analysis of Coulomb, log, and Riesz gases in arbitrary dimensions. These systems of particles with singular long-range interactions are classical ensembles of statistical mechanics and have strong connections to random matrix theory, approximation theory, statistical sampling, and random point processes. Situated at the crossroads of analysis, probability, and statistical physics, this text serves both as an introduction and as a reference for researchers, offering a self-contained presentation of the core concepts together with recent advances from the electrostatic-based approach developed by the author and her collaborators. Familiarity with standard analysis, measure theory, partial differential equations, and basic probability-at the level of a first- or second-year graduate course-is assumed. The opening chapter examines contexts in which such gases arise and provides key references. Essential results from potential theory are then introduced and related to the gases' mean-field behavior. Subsequent chapters develop the notion of electric, or modulated, energy, exploring its properties and applications to concentration and dynamical results. With its primary emphasis on the Coulomb case, the rest of the text establishes local laws, central limit theorems for fluctuations in one and two dimensions, free energy expansions, and large deviation principles for empirical fields that characterize the limiting point processes, connecting these results with crystallization problems of number-theoretic nature.