Conditional expectations of random holomorphic fields on Riemann surfaces

Abstract: We study two conditional expectations: the expected density of critical points of Gaussian random holomorphic sections of powers of a positive holomorphic line bundle over Riemann surfaces given that the random sections vanish at a point and the expected density of zeros given that the random sections has a fixed critical point. The main result is that these two rescaling conditional expectations have universal limits as the power of the line bundle tends to infinity. We will see that the short distance 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 are the same.