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 $n\to \infty$ of the form $$\exp\Big( c_1 n^2 + c_2 n\log n + c_3 n + c_4 \sqrt n + c_5 \log n + c_6 + \mathcal{O}(n^{-\frac1{12}}) \Big)$$ 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.