Chaotic analytic zero points: Exact statistics for those of a random spin state

A natural statistical ensemble of 2J points on the unit sphere can be associated, via the Majorana representation, with a random quantum state of spin J, and an exact expression is obtained here for the general k point correlation function rho(k) in this ensemble. The pair correlation rho(2) in the large-J limit takes the relatively simple form (J/2 pi)(2)g(root J/29) where g(r) = [(sinh(2) r(2) + r(4)) cosh r(2) - 2r(2) sinh(2)]/sinh(3) r(2) and theta theta is the angular separation of the pair of points on the sphere. It appears (from the numerical work of others) that, in this limit, these statistics are typical of the zero points of analytic functions associated with chaotic quantum dynamical systems.