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

TL;DR

This paper derives an asymptotic expansion of the free energy for the 2D Coulomb plasma using a quasi-free approximation and proves that linear statistics fluctuate according to a Gaussian free field at positive temperature.

Contribution

It introduces a novel asymptotic expansion of the free energy and establishes a central limit theorem for fluctuations using advanced analytical techniques.

Findings

Asymptotic free energy expansion with error bounds

Gaussian free field fluctuations of linear statistics

Rigidity bounds on local density fluctuations

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 $\kappa > 0$. 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.