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We study generalized Hermite polynomials with rectangular matrix argumentsarising in multivariate statistical analysis and the theory of zonalpolynomials. We show that these are well-suited for expressing the Wiener-Itochaos expansion of functionals of the spectral measure associated with Gaussianmatrices. In particular, we obtain the Wiener chaos expansion of Gaussiandeterminants of the form $\det(XX^T)^{1/2}$ and prove that, in the settingwhere the rows of $X$ are i.i.d. centred Gaussian vectors with a givencovariance matrix, its projection coefficients admit a geometric interpretationin terms of intrinsic volumes of ellipsoids, thus extending the content ofKabluchko and Zaporozhets (2012) to arbitrary chaotic projection coefficients.Our proofs are based on a crucial relation between generalized Hermitepolynomials and generalized Laguerre polynomials. In a second part, weintroduce the matrix analog of the classical Mehler's formula for theOrnstein-Uhlenbeck semigroup and prove that matrix-variate Hermite polynomialsare eigenfunctions of these operators. As a byproduct, we derive anorthogonality relation for Hermite polynomials evaluated at correlated Gaussianmatrices. We apply our results to vectors of independent arithmetic randomwaves on the three-torus, proving in particular a CLT in the high-energy regimefor a generalized notion of total variation on the full torus.

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