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Local microscopic behavior for 2D Coulomb gases. (English) Zbl 1379.82004

Summary: The study of two-dimensional Coulomb gases lies at the interface of statistical physics and non-Hermitian random matrix theory. In this paper we give a large deviation principle (LDP) for the empirical fields obtained, under the canonical Gibbs measure, by zooming around a point in the bulk of the equilibrium measure, up to the finest averaging scale \(N^{-1/2 + \varepsilon }\). The rate function is given by the sum of the “renormalized energy” of S. Serfaty [Coulomb gases and Ginzburg-Landau vortices. Zürich: European Mathematical Society (EMS) (2015; Zbl 1335.82002)] weighted by the inverse temperature, and of the specific relative entropy. We deduce a local law which quantifies the convergence of the empirical measures of the particles to the equilibrium measure, up to the finest scale.

MSC:

82B05 Classical equilibrium statistical mechanics (general)
60F10 Large deviations
49S05 Variational principles of physics
15B52 Random matrices (algebraic aspects)

Citations:

Zbl 1335.82002
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