Linear statistics in the Coulomb gas: cumulants beyond rotational invariance

Abstract

We consider a classical Coulomb gas of $N$ particles in dimension $d\geq2$, confined by a general external potential $U$ in the large-$N$ limit. In that limit the density of particles has a compact support called the droplet. We study smooth linear statistics ${\mathcal L}_N=\sum_{i=1}^N f({\bf x}_i)$ at large $N$. While the leading variance of ${\mathcal L}_N$ is by now well understood in many settings, much less is known about higher cumulants. When $f({\bf x})=f(|{\bf x}|)$ and the potential is rotationally invariant, these cumulants were computed recently at leading order. We derive explicit formulas for the leading third and fourth cumulants for general smooth droplets, without assuming that either the potential or the test function is radial. The answers are expressed as intrinsic boundary integrals over the equilibrium droplet. The third cumulant depends on the cube of the jump of the normal derivative of the harmonic extension of $f$, while the fourth cumulant also involves the exterior Dirichlet-to-Neumann operator, the mean curvature of the boundary, and normal derivatives of the background density. Our derivation uses the tilted potential $U_s=U+s f/β$, the associated deformation of the droplet, and shape derivatives of the Dirichlet energy. This deformation has a natural interpretation as the infinitesimal motion of the free boundary in the obstacle-problem formulation of the equilibrium measure. Our formulas extend previous studies by Wiegmann and Zabrodin in two dimensions for $β=2$. In their setting, we provide a rigorous proof of the limiting third and fourth cumulants, supporting the proposed formulas for general Coulomb gases. We specialize the general formulas to the disk, the half-plane and the sphere, and check them against exactly solvable non-radial deformations, including translated elliptic droplets in $d=2$ and spheroidal deformations in $d=3$.