  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.
