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

We study large $n$ expansions for the partition function of a Coulomb gas $$Z_n=\frac{1}{{\pi }^n}{\int }_{{\mathbb{C}}^n}\prod _{1\le i<j\le n}\mathrm{\mid }{z}_i-z_j{\mathrm{\mid }}^2\prod _{i=1}^n{e}^{-nQ(z_i)}{d}^2{z}_i,$$ 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 $C_1,\dots ,C_5$ are certain geometric functionals. The $n$-dependent term ${\mathcal{G}}_n$ is bounded as $n\to \mathrm{\infty }$; 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.