Derivative principles for invariant ensembles

Abstract

A big class of ensembles of random matrices are those which are invariant under group actions so that the eigenvectors become statistically independent of the eigenvalues. In the present work we show that this is not the sole consequence but that the joint probability distribution of the eigenvalues can be expressed in terms of a differential operator acting on the distribution of some other matrix quantities which are sometimes easier accessible. Those quantities might be the diagonal matrix entries as it is the case for Hermitian matrices or some pseudo-diagonal entries for other matrix spaces. These representations are called derivative principles. We show them for the additive spaces of the Hermitian, Hermitian antisymmetric, Hermitian anti-self-dual, and complex rectangular matrices as well as for the two multiplicative matrix spaces of the positive definite Hermitian matrices and of the unitary matrices. In all six cases we prove a uniqueness of the derivative principle as it is not trivial to see that the derivative operator has a trivial kernel on the selected spaces of the identified matrix quantities.

Stable distributions and domains of attraction for unitarily invariant Hermitian random matrix ensembles

“…Indeed it seems to be rather straightforward to three of those, namely random matrices invariant under their respective conjugate group actions that are imaginary anti-symmetric, Hermitian anti-self-dual, and Hermitian chiral. The reason is that there are derivative principal for all three classes [57]. Those can be traced back to known group integrals.…”

“…It is proven in subsection 2.3. So far this theorem was only shown [29,38,64,57] when a density for the measure exists; it is restated in Proposition 7.…”

“…[29,64]). These are summarised and stated as an existence and uniqueness theorem in [57,Cor 3.4], with an analytical conditions of F assumed for technical purposes.…”

“…To circumvent this disadvantage, we exploit a one-to-one relation between the joint probability distribution of the eigenvalue and the one of the diagonal entries of a unitarily invariant Hermitian random matrix. This relation is known as a derivative principle [29,38,64,57]. The idea is that the diagonal entries again build a vector space with respect matrix addition so that the well-known additive convolution formula for vector spaces and the corresponding Fourier transform can be applied.…”

Spectral statistics for the difference of two Wishart matrices

“…This category, in turn, belongs to that of Muttalib-Borodin biorthogonal ensembles [77,78]. The second approach, on the other hand, relies on the application of the so called derivative principle which gives a very interesting and powerful connection between the joint probability density of eigenvalues of a random matrix drawn from a unitarily invariant ensemble and that of its diagonal elements [71,79,80]. This leads to the realization of a Pólya ensemble, i.e., polynomial ensemble of a derivative type [22,23,76].…”