Conditional Expectations of Random Holomorphic Fields on Riemann Surfaces

We study two conditional expectations: K-n (z vertical bar p) of the expected density of critical points of Gaussian random holomorphic sections s(n) is an element of H-0(M, L-n) of powers of a positive holomorphic line bundle (L, h) over Riemann surfaces (M,omega) given that the random sections vanish at a point and D-n (z vertical bar q) of the expected density of zeros given that the random sections have a fixed critical point. The critical points are points del(h)ns(n) = 0 where. del(h)n is the smooth Chern connection of the Hermitian metric h(n). The main result is that the rescaling conditional expectations K-n (p + u/root n vertical bar p) and Dn (q + u/root n vertical bar q) have universal limits K-infinity (u vertical bar 0) and D-infinity (v vertical bar 0) as the power of the line bundle tends to infinity. We will see that the short-distance behaviors of these two conditional expectations are quite different: the behavior between critical points and the conditioning zero is neutral while there is a repulsion between zeros and the conditioning critical point. But the long-distance behaviors of these two rescaling densities are the same.SCI(E)ARTICLE144406-443