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# Advances in Theoretical and Mathematical Physics

## Volume 23 (2019)

### Number 4

### The two-dimensional Coulomb plasma: quasi-free approximation and central limit theorem

Pages: 841 – 1002

DOI: https://dx.doi.org/10.4310/ATMP.2019.v23.n4.a1

#### Authors

#### Abstract

For the two-dimensional one-component Coulomb plasma, we derive an asymptotic expansion of the free energy up to order $N$, the number of particles of the gas, with an effective error bound $N^{1-\kappa}$ for some constant $\kappa \gt 0$. This expansion is based on approximating the Coulomb gas by a quasi-free Yukawa gas. Further, we prove that the fluctuations of the linear statistics are given by a Gaussian free field at any positive temperature. Our proof of this central limit theorem uses a loop equation for the Coulomb gas, the free energy asymptotics, and rigidity bounds on the local density fluctuations of the Coulomb gas, which we obtained in a previous paper.

R. Bauerschmidt and P. Bourgade were partially supported by NSF grant DMS-1513587.

M. Nikula and H.-T. Yau were partially supported by NSF grants DMS-1606305 and 1855509, and by a Simons Investigator award.

Published 16 January 2020