Correlations between zeros and critical points of random analytic functions

TL;DR

This paper investigates the universal behavior of the correlation between zeros and critical points of Gaussian random holomorphic sections on Kähler manifolds, revealing a universal limit and local repulsion and neutrality phenomena.

Contribution

It establishes the universal rescaling limit of zero-critical point correlations and characterizes their short and long-range interactions.

Findings

Universal rescaling limit matches the correlation in Bargmann-Fock space.

Short-range interactions show repulsion between zeros and critical points.

Long-range interactions exhibit neutrality, with no significant correlation.

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 K\"ahler 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, there is a 'repulsion' between zeros and critical points for the short range; and a 'neutrality' for the long range.