The two-dimensional Coulomb gas: fluctuations through a spectral gap

We study a class of radially symmetric Coulomb gas ensembles at inverse temperature $\beta=2$, for which the droplet consists of a number of concentric annuli, having at least one bounded ``gap'' $G$, i.e., a connected component of the complement of the droplet, which disconnects the droplet. Let $n$ be the total number of particles. Among other things, we deduce fine asymptotics as $n \to \infty$ for the edge density and the correlation kernel near the gap, as well as for the cumulant generating function of fluctuations of smooth linear statistics. We typically find an oscillatory behaviour in the distribution of particles which fall near the edge of the gap. These oscillations are given explicitly in terms of a discrete Gaussian distribution, weighted Szegő kernels, and the Jacobi theta function, which depend on the parameter $n$.