Correlations between zeros and critical points of random analytic functions

Abstract:

We study the two-point correlation $K^m_n(z,w)$ between zeros and critical points of Gaussian random holomorphic sections $s_n$ over Kähler manifolds. The critical points are points $\nabla _{h^n} s_n=0$, where $\nabla _{h^n}$ is the smooth Chern connection with respect to the Hermitian metric $h^n$ on line bundle $L^n$. The main result is that the rescaling limit of $K^m_n(z_0+\frac u{\sqrt n}, z_0+\frac v{\sqrt n})$ for any $z_0\in M$ is universal as $n$ tends to infinity. In fact, the universal rescaling limit is the two-point correlation between zeros and critical points of Gaussian analytic functions for the Bargmann–Fock space of level $1$. Furthermore, in the length scale of order $n^{-\frac 12}$, there is a “repulsion" between zeros and critical points for the short range, and a “neutrality" for the long range.