Free energy and fluctuations in the random normal matrix model with spectral gaps

TL;DR

This paper analyzes large n expansions of the partition function for a Coulomb gas with complex potential, including spectral gaps and singularities, and studies the resulting particle fluctuations and outpost distributions.

Contribution

It provides a detailed large n expansion of the free energy for Coulomb gases with complex geometries and spectral gaps, and characterizes particle fluctuation distributions.

Findings

Large n expansion of free energy with geometric functionals

Fluctuations approximated by Gaussian plus displacement terms

Number of particles near outposts converges to Heine distribution

Abstract

We study large $n$ expansions for the partition function of a Coulomb gas $$Z_n=\frac 1 {\pi^n}\int_{\mathbb{C}^n}\prod_{1\le i<j\le n}|z_i-z_j|^2\prod_{i=1}^n e^{-nQ(z_i)}\, d^2 z_i,$$ where $Q$ is a radially symmetric confining potential on the complex plane $\mathbb{C}$. The droplet is not assumed to be connected, but may consist of a number of disjoint connected annuli and possibly a central disk. The boundary condition is ``soft edge'', i.e., $Q$ is smooth in a $\mathbb{C}$-neighbourhood of the droplet. We include the following possibilities: (i) existence of ``outposts'', i.e., components of the coincidence set which falls outside of the droplet, (ii) a conical (or Fisher-Hartwig) singularity at the origin, (iii) perturbations $Q-\frac h n$ where $h$ is a smooth radially symmetric test-function. In each case, the free energy $\log Z_n$ admits a large $n$ expansion of the form \begin{equation*}\log Z_n=C_1n^2+C_2n\log n+C_3 n+C_4\log n+C_5+\mathcal{G}_{n}+o(1)\end{equation*} where $C_1,\ldots,C_5$ are certain geometric functionals. The $n$-dependent term $\mathcal{G}_n$ is bounded as $n\to\infty$; it arises in the presence of spectral gaps. We use the free energy expansions to study the distribution of fluctuations of linear statistics. We prove that the fluctuations are well approximated by the sum of a Gaussian and certain independent terms which provide the displacement of particles from one component to another. This displacement depends on $n$ and is expressed in terms of the Heine distribution. We also prove (under suitable assumptions) that the number of particles which fall near a spectral outpost converges to a Heine distribution.