Fluctuations of Two Dimensional Coulomb Gases

• Published in: Geometric and functional analysis Vol. 28; no. 2; pp. 443 - 508
• Main Authors: Leblé, Thomas, Serfaty, Sylvia
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.04.2018; Springer Nature B.V
• Subjects: Analysis; Confining; Convergence; Mathematics; Mathematics and Statistics; Upper bounds; Variation; Linear statistics; 82B05; Log gas; 60B10; 60B20; Ginibre ensemble; Gaussian free field; Coulomb gas; Central Limit Theorem; 60K35; 60F05; 60G15; ensembles
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-018-0443-1

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.