Spherical Functions Approach to Sums of Random Hermitian Matrices

Abstract

We present an approach to sums of random Hermitian matrices via the theory of spherical functions for the Gelfand pair (U(n) ⋉ Herm(n), U(n)). It is inspired by a similar approach of Kieburg and Kösters for products of random matrices. The spherical functions have determinantal expressions because of the Harish-Chandra/ItzyksonZuber integral formula. It leads to remarkably simple expressions for the spherical transform and its inverse. The spherical transform is applied to sums of unitarily invariant random matrices from polynomial ensembles and the subclass of polynomial ensembles of derivative type (in the additive sense), which turns out to be closed under addition. We finally present additional detailed calculations for the sum with a random matrix from a Laguerre Unitary Ensemble.

Multiplicative convolution of real asymmetric and real anti-symmetric matrices

“…A particular limit of Theorem 3.3 is of special interest, namely when a → 1 1 n . Then we obtain a jPDF which resembles those for Pólya ensembles on Gl C (n), see [33,20,27]. Indeed this result shows that the weight Aσ(e y ) with y ∈ R has to be Pólya function [40,41,42], for the same reasons as discussed in [30].…”

“…We constructed a theoretical basis for dealing with the multiplicative convolution corresponding to the action of the general linear group G = Gl R (2n) on the even dimensional real antisymmetric matrices H = o(2n), i.e. (g, x) → gxg T with g ∈ G and x ∈ H. This approach is based on harmonic analysis [23] and follows the same ideas as already employed for various convolutions of the additive [33,27] and the multiplicative [29,30] type. The only requirement to apply these ideas is that the probability density P H (x) has to be K-invariant, i.e.…”

“…However the spherical transforms of arbitrary distributions P G and P H can have very complicated expressions. As we already know from the multiplicative convolution of complex matrices [29] and from the additive convolution [33,20] of Hermitian antisymmetric matrices, Hermitian matrices, Hermitian anti-self-dual matrices, and complex rectangular matrices one has to restrict the class of ensembles for the results to be analytically tractable. For this reason we define the two sets of functions…”

Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

“…In §3 we derive Theorem 1.1, and its analogue in the odd dimensional case, as well as for the matrix structure relating to (1.11) in the case of the unitary symplectic group. In the final subsection we give a technically different derivation of Theorem 1.1, which in keeping with the recent works [28,29,31] highlights the role of matrix transforms, although again the matrix integral (1.3) plays a key role. Special cases, including the result Corollary 1.2 are given in §4 for the even dimensional case, and in §5 for the odd dimensional case.…”

“…Kieburg and Kösters have recently introduced the matrix spherical transform to the analysis of both the eigenvalue and singular value distribution of products of unitary invariant matrix ensembles [28,29,19,27]. It was pointed out by Kuijlaars and Roman [31] that the analogue in the case of sums of unitary invariant matrix ensembles is the matrix Fourier transform. The latter is equivalent to the matrix-valued Fourier-Laplace transform (3.4).…”