Conditional Expectations of Random Holomorphic Fields on Riemann Surfaces

Abstract

We study two conditional expectations: |$K_n(z| p)$| of the expected density of critical points of Gaussian random holomorphic sections |$s_n \in {H^0(M,L^n)}$| 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 |$D_n(z|q)$| of the expected density of zeros given that the random sections have a fixed critical point. The critical points are points |$\nabla_{h^n} s_n=0$| where |$\nabla_{h^n}$| is the smooth Chern connection of the Hermitian metric |$h^n$|⁠. The main result is that the rescaling conditional expectations |$K_n(p+\frac u{\sqrt n}|p)$| and |$D_n(q+\frac u{\sqrt n}|q)$| have universal limits |$K_\infty(u|0)$| and |$D_\infty(v|0)$| 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.