Asymptotics of the real eigenvalue distribution for the real spherical ensemble

The real Ginibre spherical ensemble consists of random matrices of the form $A B^{-1}$, where $A,B$ are independent standard real Gaussian $N \times N$ matrices. The expected number of real eigenvalues is known to be of order $\sqrt{N}$. We consider the probability $p_{N.M}^{\rm r}$ that there are $M$ real eigenvalues in various regimes. These are when $M$ is proportional to $N$ (large deviations), when $N$ is proportional to $\sqrt{N}$ (intermediate deviations), and when $M$ is in the neighbourhood of the mean (local central limit theorem). This is done using a Coulomb gas formalism in the large deviations case, and by determining the leading asymptotic form of the generating function for the probabilities in the case of intermediate deviations (the local central limit regime was known from earlier work). Moreover a matching of the left tail asymptotics of the intermediate deviation regime with that of the right tail of the large deviation regime is exhibited, as is a matching of the right tail intermediate deviation regime with the leading order form of the probabilities in the local central limit regime. We also give the leading asymptotic form of $p_{N,0}^{\rm r}$, i.e. the probability of no real eigenvalues.