Large deviations for the empirical measure of random polynomials: revisit of the Zeitouni-Zelditch theorem

TL;DR

This paper revisits and extends large deviation principles for empirical measures of random orthogonal polynomials with Gaussian coefficients, focusing on Kac and elliptic polynomials, and broadens the scope from complex to real coefficients.

Contribution

It extends the Zeitouni-Zelditch theorem to real Gaussian coefficients and simplifies the approach by removing geometric prerequisites.

Findings

Large deviation principles hold for real Gaussian coefficients.

Results apply to classical Kac and elliptic polynomials.

No geometric knowledge required for the proofs.

Abstract

This article revisits the work by Ofer Zeitouni and Steve Zelditch on large deviations for the empirical measures of random orthogonal polynomials with i.i.d. Gaussian complex coefficients, and extends this result to real Gaussian coefficients. This article does not require any knowledge in geometry. For clarity, we focus on two classical cases: Kac polynomials and elliptic polynomials.