Geometric and Functional Analysis

, Volume 28, Issue 2, pp 443–508 | Cite as

Fluctuations of Two Dimensional Coulomb Gases

  • Thomas Leblé
  • Sylvia Serfaty


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.

Mathematics Subject Classification

60F05 60K35 60B10 60B20 82B05 60G15 

Keywords and phrases

Coulomb gas β-ensembles Log gas Central Limit Theorem Gaussian free field Linear statistics Ginibre ensemble 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Sorbonne Universités, UPMC Univ. Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis LionsParisFrance
  2. 2.Courant Institute, New York UniversityNew YorkUSA

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