  For the plane, sphere, and hyperbolic plane we consider the canonical
invariant determinantal point processes with intensity rho dnu, where nu is the
corresponding invariant measure. We show that as rho converges to infinity,
after centering, these processes converge to invariant H1 noise. More
precisely, for all functions f in the interesection of H1(nu) and L1(nu) the
distribution of sum f(z) - rho/pi integral f dnu converges to Gaussian with
mean 0 and variance given by ||f||_H1^2 / (4 pi).
