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We study large $n$ expansions for the partition function of a Coulomb gas$$Z_n=\frac 1 {\pi^n}\int_{\mathbb{C}^n}\prod_{1\le i<j\len}|z_i-z_j|^2\prod_{i=1}^n e^{-nQ(z_i)}\, d^2 z_i,$$ where $Q$ is a radiallysymmetric confining potential on the complex plane $\mathbb{C}$.  The droplet is not assumed to be connected, but may consist of a number ofdisjoint connected annuli and possibly a central disk. The boundary conditionis ``soft edge'', i.e., $Q$ is smooth in a $\mathbb{C}$-neighbourhood of thedroplet.  We include the following possibilities: (i) existence of ``outposts'', i.e.,components of the coincidence set which falls outside of the droplet, (ii) aconical (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 theform \begin{equation*}\log Z_n=C_1n^2+C_2n\log n+C_3 n+C_4\logn+C_5+\mathcal{G}_{n}+o(1)\end{equation*} where $C_1,\ldots,C_5$ are certaingeometric 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 fluctuationsof linear statistics. We prove that the fluctuations are well approximated bythe sum of a Gaussian and certain independent terms which provide thedisplacement of particles from one component to another. This displacementdepends on $n$ and is expressed in terms of the Heine distribution. We alsoprove (under suitable assumptions) that the number of particles which fall neara spectral outpost converges to a Heine distribution.

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