Matrix Hermite polynomials, Random determinants and the geometry of Gaussian fields

TL;DR

This paper develops matrix Hermite polynomials for Gaussian matrices, deriving their chaos expansion, geometric interpretations, and eigenfunction properties, with applications to Gaussian determinants and random waves.

Contribution

It introduces matrix Hermite polynomials and their properties, extending classical results to matrix-variate cases and linking them to geometric and spectral analysis.

Findings

Wiener chaos expansion of Gaussian determinants obtained.

Projection coefficients linked to geometric volumes of ellipsoids.

Eigenfunction property of matrix Hermite polynomials for Ornstein-Uhlenbeck operators.

Abstract

We study generalized Hermite polynomials with rectangular matrix arguments arising in multivariate statistical analysis and the theory of zonal polynomials. We show that these are well-suited for expressing the Wiener-Ito chaos expansion of functionals of the spectral measure associated with Gaussian matrices. In particular, we obtain the Wiener chaos expansion of Gaussian determinants of the form $\det(XX^T)^{1/2}$ and prove that, in the setting where the rows of $X$ are i.i.d. centred Gaussian vectors with a given covariance matrix, its projection coefficients admit a geometric interpretation in terms of intrinsic volumes of ellipsoids, thus extending the content of Kabluchko and Zaporozhets (2012) to arbitrary chaotic projection coefficients. Our proofs are based on a crucial relation between generalized Hermite polynomials and generalized Laguerre polynomials. In a second part, we introduce the matrix analog of the classical Mehler's formula for the Ornstein-Uhlenbeck semigroup and prove that matrix-variate Hermite polynomials are eigenfunctions of these operators. As a byproduct, we derive an orthogonality relation for Hermite polynomials evaluated at correlated Gaussian matrices. We apply our results to vectors of independent arithmetic random waves on the three-torus, proving in particular a CLT in the high-energy regime for a generalized notion of total variation on the full torus.