Large gap probabilities of complex and symplectic spherical ensembles with point charges

We consider $n$ eigenvalues of complex and symplectic induced spherical ensembles, which can be realised as two-dimensional determinantal and Pfaffian Coulomb gases on the Riemann sphere under the insertion of point charges. For both cases, we show that the probability that there are no eigenvalues in a spherical cap around the poles has an asymptotic behaviour as $n\to \mathrm{\infty }$ of the form $$\exp (c_1{n}^2+c_{2}n\log n+c_{3}n+c_4\sqrt{n}+c_5\log n+c_6+\mathcal{O}(n^{-\frac{1}{12}}))$$ and determine the coefficients explicitly. Our results provide the second example of precise (up to and including the constant term) large gap asymptotic behaviours for two-dimensional point processes, following a recent breakthrough by Charlier.