Partition functions of two-dimensional Coulomb gases with circular root- and jump-type singularities

In this paper, we study the random polynomial $p_n(\rho):=\prod_{j=1}^n (|z_j|-\rho)$, where the points $\{z_j\}_{j=1}^n$ are the eigenvalue moduli of random normal matrices with a radially symmetric potential. We establish precise large $n$ asymptotic expansions for the moment generating function $$\mathbb{E}\!\left[e^{\tfrac{u}{\pi}\mathrm{Im}\log p_n(\rho)}\, e^{a\,\mathrm{Re}\log p_n(\rho)}\right], \qquad u\in\mathbb{R}, \; a>-1,$$ where $\rho>0$ lies in the bulk of the spectral droplet. The asymptotic expansion is expressed in terms of parabolic cylinder functions, which confirms a conjecture of Byun and Charlier. This also provides the first free energy expansion of two-dimensional Coulomb gases with general circular root- and jump-type singularities. While the $a=0$ case has already been widely studied in the literature due to its relation to counting statistics, we also obtain new results for this special case.