Matrix Hermite polynomials, Random determinants and the geometry of Gaussian fields
Annales Henri Lebesgue, Volume 6 (2023), pp. 975-1030.


Keywords Generalized Hermite polynomials, Gaussian random matrices, Zonal polynomials, Wiener chaos expansions, Intrinsic and mixed volumes, Arithmetic Random Waves, Limit Theorems


We study generalized Hermite polynomials with rectangular matrix arguments arising in multivariate statistical analysis. We argue that these are well-suited for expressing the Wiener–Itô chaos expansion of functionals of the spectral measure associated with Gaussian matrices. More specifically, 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. Gaussian vectors, its projection coefficients admit a geometric interpretation in terms of intrinsic volumes of ellipsoids, thus extending the framework of Kabluchko and Zaporozhets (2012). Our proofs rely on a crucial relation between Hermite polynomials and Laguerre polynomials. We introduce the matrix analog of the classical Mehler’s formula for the Ornstein-Uhlenbeck semigroup and prove that matrix Hermite polynomials are eigenfunctions of these operators. We apply our results to the asymptotic study of a total variation associated with vectors of Arithmetic Random Waves on the full three-torus.


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