Correlation functions for zeros of a Gaussian power series and Pfaffians

TL;DR

This paper demonstrates that zeros of a Gaussian power series form a Pfaffian point process and that related moments can also be expressed via Pfaffians, revealing new structural insights into these random zeros.

Contribution

It establishes the Pfaffian structure of zeros of Gaussian power series and extends Pfaffian representations to product moments of their absolute values and signatures.

Findings

Zeros form a Pfaffian point process

Product moments can be expressed by Pfaffians

Provides new structural understanding of Gaussian power series zeros

Abstract

We show that the zeros of the random power series with i.i.d. real Gaussian coefficients form a Pfaffian point process. We further show that the product moments for absolute values and signatures of the power series can also be expressed by Pfaffians.