Hole probabilities and balayage of measures for planar Coulomb gases

We study hole probabilities of two-dimensional Coulomb gases with a general potential and arbitrary temperature. The hole region $U$ is assumed to satisfy $\partial U\subset S$, where $S$ is the support of the equilibrium measure $\mu$. Let $n$ be the number of points. As $n \to \infty$, we prove that the probability that no points lie in $U$ behaves like $\exp(-Cn^{2}+o(n^{2}))$. We determine $C$ in terms of $\mu$ and the balayage measure $\nu = \mathrm{Bal}(\mu|_{U},\partial U)$. If $U$ is unbounded, then $C$ also involves the Green function of $\Omega$ with pole at $\infty$, where $\Omega$ is the unbounded component of $U$. We also provide several examples where $\nu$ and $C$ admit explicit expressions: we consider several point processes, such as the elliptic Ginibre, Mittag-Leffler, and spherical point processes, and various hole regions, such as circular sectors, ellipses, rectangles, and the complement of an ellipse. This work generalizes previous results of Adhikari and Reddy in several directions.