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



Roland Bauerschmidt (Statistical Laboratory, DPMMS, University of Cambridge, United Kingdom)

Paul Bourgade (Department of Mathematics, New York University, New York, N.Y., U.S.A.)

Miika Nikula (Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, U.S.A.)

Horng-Tzer Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)


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