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→ ∞ of the form exp( c₁ n² + c₂ nlog n + c₃ n + c₄ √ n + c₅ log n + c₆ + O(n^-112) ) 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.