The Ginibre ensemble and Gaussian analytic functions

TL;DR

This paper links the characteristic polynomial of Ginibre matrices to Gaussian analytic functions through a recursive process, revealing insights into their zero distributions and providing explicit formulas for the limiting functions.

Contribution

It introduces a recursive description of characteristic polynomials leading to a new class of Gaussian analytic functions as limits, connecting random matrix theory and complex analysis.

Findings

Zeros of Gaussian analytic functions and Ginibre ensemble show similar local repulsion.

Derived explicit formulas for the limiting Gaussian analytic functions.

Established a recursive process analogous to Pólya's urn for matrix characteristic polynomials.

Abstract

We show that as $n$ changes, the characteristic polynomial of the $n\times n$ random matrix with i.i.d. complex Gaussian entries can be described recursively through a process analogous to P\'olya's urn scheme. As a result, we get a random analytic function in the limit, which is given by a mixture of Gaussian analytic functions. This gives another reason why the zeros of Gaussian analytic functions and the Ginibre ensemble exhibit similar local repulsion, but different global behavior. Our approach gives new explicit formulas for the limiting analytic function.