Average Discrete Energies of Spectra of Gaussian Random Matrices

Abstract

We present exact formulas for the averages of various discrete energies of certain point processes in the plane and two-dimensional sphere. Specifically, we consider point processes defined by the spectra of gaussian random matrices and their inverse images under stereographic projection on the Riemann sphere. The main attention is given to the discrete Coulomb energy, discrete logarithmic energy, and their analogs involving geodesic distances on the Riemann sphere. It is shown that the average discrete energies are expressed by integrals of certain special type taken over the Riemann sphere. This enables us to estimate their values and asymptotics as the size of random matrix tends to infinity. Analogous results are obtained for the three-point energy function considered by M. Atiyah. We also present several related conjectures and possible generalizations.