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Fluctuations of Two Dimensional Coulomb Gases

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Abstract

We prove a Central Limit Theorem for the linear statistics of two-dimensional Coulomb gases, with arbitrary inverse temperature and general confining potential, at the macroscopic and mesoscopic scales and possibly near the boundary of the support of the equilibrium measure. This can be stated in terms of convergence of the random electrostatic potential to a Gaussian Free Field.

Our result is the first to be valid at arbitrary temperature and at the mesoscopic scales, and we recover previous results of Ameur-Hendenmalm-Makarov and Rider-Virág concerning the determinantal case, with weaker assumptions near the boundary. We also prove moderate deviations upper bounds, or rigidity estimates, for the linear statistics and a convergence result for those corresponding to energy-minimizers.

The method relies on a change of variables, a perturbative expansion of the energy, and the comparison of partition functions deduced from our previous work. Near the boundary, we use recent quantitative stability estimates on the solutions to the obstacle problem obtained by Serra and the second author.

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Author notes

  1. Thomas Leblé

    Present address: Courant Institute, New York University, 251 Mercer St, New York, NY, 10012, USA

Authors and Affiliations

  1. Sorbonne Universités, UPMC Univ. Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu, 75005, Paris, France

    Thomas Leblé & Sylvia Serfaty

  2. Courant Institute, New York University, 251 Mercer St, New York, NY, 10012, USA

    Sylvia Serfaty

Authors
  1. Thomas Leblé
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  2. Sylvia Serfaty
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Correspondence to Sylvia Serfaty.

Additional information

We use \(\frac{\beta}{2}\) instead of β in order to match the normalizations in the existing literature. The first sum in (1.2) is twice the physical one, but it is more convenient for our analysis.

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Leblé, T., Serfaty, S. Fluctuations of Two Dimensional Coulomb Gases. Geom. Funct. Anal. 28, 443–508 (2018). https://doi.org/10.1007/s00039-018-0443-1

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