Gaussian Analytic functions in the polydisk

TL;DR

This paper investigates the behavior of hyperbolic Gaussian analytic functions in the polydisk, analyzing fluctuations, large deviations, and hole probabilities, extending methods from one-dimensional cases to higher dimensions.

Contribution

It extends the analysis of Gaussian analytic functions to the polydisk, providing asymptotic fluctuation results and large deviation estimates in multiple complex dimensions.

Findings

Asymptotic behavior of fluctuations as intensities grow

Probability estimates for large deviations of linear statistics

Proof of a hole theorem in the polydisk setting

Abstract

We study hyperbolic Gaussian analytic functions in the unit polydisk of $\mathbb C^n$. Following the scheme previously used in the unit ball we first study the asymptotics of fluctuations of linear statistics as the directional intensities $L_j$, $j=1,\dots,n$ tend to $\infty$. Then we estimate the probability of large deviations of such linear statistics and use the estimate to prove a hole theorem. Our proofs are inspired by the methods of M. Sodin and B. Tsirelson for the one-dimensional case, and B. Shiffman and S. Zelditch for the study of the analogous problem for compact K\"ahler manifolds.