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Local microscopic behavior for 2D Coulomb gases

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Abstract

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 Serfaty et al. 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.

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Acknowledgments

The author would like to thank his Ph.D. supervisor, Sylvia Serfaty, for helpful comments on this work, as well as the anonymous referees for many useful corrections and comments.

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Authors and Affiliations

  1. UMR 7598, Laboratoire Jacques-Louis Lions, Sorbonne Universités, UPMC Univ. Paris 06, CNRS, 75005, Paris, France

    Thomas Leblé

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  1. Thomas Leblé
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Correspondence to Thomas Leblé.

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Leblé, T. Local microscopic behavior for 2D Coulomb gases. Probab. Theory Relat. Fields 169, 931–976 (2017). https://doi.org/10.1007/s00440-016-0744-y

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