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Portugaliae Mathematica

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Volume 73, Issue 4, 2016, pp. 247–278
DOI: 10.4171/PM/1987

Published online: 2016-11-30

Large systems with Coulomb interactions: Variational study and statistical mechanics

Sylvia Serfaty[1]

(1) Université Pierre et Marie Curie (Paris VI), France

Systems with Coulomb and logarithmic interactions arise in various settings: an instance is the classical Coulomb gas which in some cases happens to be a random matrix ensemble, another is vortices in the Ginzburg-Landau model of superconductivity, where one observes in certain regimes the emergence of densely packed point vortices forming perfect triangular lattice patterns named Abrikosov lattices, a third is the study of Fekete points which arise in approximation theory. In this review, we describe tools to study such systems and derive a next order (beyond mean field limit) ‘‘renormalized energy’’ that governs microscopic patterns of points. We present the derivation of the limiting problem and the question of its minimization and its link with the Abrikosov lattice and crystallization questions. We also discuss generalizations to Riesz interaction energies and the statistical mechanics of such systems. This is based on joint works with Etienne Sandier, Nicolas Rougerie, Simona Rota Nodari, Mircea Petrache, and Thomas Leblé.

Keywords: Coulomb systems, Coulomb gases, log gases, Large Deviations Principle, Abrikosov lattices, Fekete points, random matrices

Serfaty Sylvia: Large systems with Coulomb interactions: Variational study and statistical mechanics. Port. Math. 73 (2016), 247-278. doi: 10.4171/PM/1987