Fluctuations in the logarithmic energy for zeros of random polynomials on the sphere

Smale’s Seventh Problem asks for an efficient algorithm to generate a configuration of n points on the sphere that nearly minimizes the logarithmic energy. As a candidate starting configuration for this problem, Armentano, Beltrán and Shub considered the set of points given by the stereographic projection of the roots of the random elliptic polynomial of degree n and computed the expected logarithmic energy. We study the fluctuations of the logarithmic energy associated to this random configuration and prove a central limit theorem. Our approach shows that all cumulants of the logarithmic energy are asymptotically linear in n, and hence the energy is well-concentrated on the scale of \(\sqrt{n}\) $\sqrt{n}$ .