  We consider canonical determinantal random point processes with N particles
on a compact Riemann surface X defined with respect to the constant curvature
metric. In the higher genus (hyperbolic) cases these point processes may be
defined in terms of automorphic forms. We establish strong exponential
concentration of measure type properties involving Dirichlet norms of linear
statistics. This gives an optimal Central Limit Theorem (CLT), saying that the
fluctuations of the corresponding empirical measures converge, in the large N
limt, towards the Laplacian of the Gaussian free field on X in the strongest
possible sense. The CLT is also shown to be equivalent to a new sharp strong
Szeg\"o type theorem for Toeplitz determinants in this context. One of the
ingredients in the proofs are new Bergman kernel asymptotics providing
exponentially small error terms in a constant curvature setting.
