Rigidity of the Peres-Virág class

Abstract

Consider the hyperbolic Gaussian analytic function (GAF) \begin{equation*} F_{\mathbb{D}}(z)=\sum_{n=0}^{\infty }ξ_{n}z^{n},\qquad |z|<1, \end{equation*} where the $ξ_{n}$ are independent standard complex Gaussian variables. Peres-Virág theorem tells that the distribution of the zeros of $F_{\mathbb{D}}(z)$ is provided by a Determinantal Point Process (DPP). We prove that $F_{\mathbb{D}}(z)$ is the only GAF whose zeros are determinantal with a Hermitian kernel, up to multiplication by a zero-free holomorphic function and a one-to-one holomorphic change of variable, with a \ non-degeneracy condition imposed to exclude trivial cases. The proof identifies a geometric property of GAF&#39;s with (local) determinantal zeros: the metric defined by the first intensity must have constant curvature $-4$.