  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.
