Zeros of the i.i.d. Gaussian Laurent series on an annulus: weighted Szeg\H{o} kernels and permanental-determinantal point processes

TL;DR

This paper investigates the zeros of a Gaussian analytic function on an annulus, revealing their structure as a permanental-determinantal point process and exploring their symmetries, invariances, and limits to simpler models.

Contribution

It introduces a new class of zero point processes linked to weighted Szeg ext{o} kernels and characterizes their properties as a PDPP with a detailed analysis of their correlation functions.

Findings

Zero set forms a permanental-determinantal point process.

The process exhibits symmetries under q-inversion and parameter transformations.

Limit cases interpolate between known determinantal point processes.

Abstract

On an annulus ${\mathbb{A}}_q :=\{z \in {\mathbb{C}}: q < |z| < 1\}$ with a fixed $q \in (0, 1)$, we study a Gaussian analytic function (GAF) and its zero set which defines a point process on ${\mathbb{A}}_q$ called the zero point process of the GAF. The GAF is defined by the i.i.d.~Gaussian Laurent series such that the covariance kernel parameterized by $r >0$ is identified with the weighted Szeg\H{o} kernel of ${\mathbb{A}}_q$ with the weight parameter $r$ studied by Mccullough and Shen. The GAF and the zero point process are rotationally invariant and have a symmetry associated with the $q$-inversion of coordinate $z \leftrightarrow q/z$ and the parameter change $r \leftrightarrow q^2/r$. When $r=q$ they are invariant under conformal transformations which preserve ${\mathbb{A}}_q$. Conditioning the GAF by adding zeros, new GAFs are induced such that the covariance kernels are also given by the weighted Szeg\H{o} kernel of Mccullough and Shen but the weight parameter $r$ is changed depending on the added zeros. We also prove that the zero point process of the GAF provides a permanental-determinantal point process (PDPP) in which each correlation function is expressed by a permanent multiplied by a determinant. Dependence on $r$ of the unfolded 2-correlation function of the PDPP is studied. If we take the limit $q \to 0$, a simpler but still non-trivial PDPP is obtained on the unit disk ${\mathbb{D}}$. We observe that the limit PDPP indexed by $r \in (0, \infty)$ can be regarded as an interpolation between the determinantal point process (DPP) on ${\mathbb{D}}$ studied by Peres and Vir\'ag ($r \to 0$) and that DPP of Peres and Vir\'ag with a deterministic zero added at the origin ($r \to \infty$).