Polynomial Ensembles and Pólya Frequency Functions

Abstract

We study several kinds of polynomial ensembles of derivative type which we propose to call Pólya ensembles. These ensembles are defined on the spaces of complex square, complex rectangular, Hermitian, Hermitian anti-symmetric and Hermitian anti-self-dual matrices, and they have nice closure properties under the multiplicative convolution for the first class and under the additive convolution for the other classes. The cases of complex square matrices and Hermitian matrices were already studied in former works. One of our goals is to unify and generalize the ideas to the other classes of matrices. Here we consider convolutions within the same class of Pólya ensembles as well as convolutions with the more general class of polynomial ensembles. Moreover, we derive some general identities for group integrals similar to the Harish-Chandra-Itzykson-Zuber integral, and we relate Pólya ensembles to Pólya frequency functions. For illustration we give a number of explicit examples for our results.

arXiv comment28 pages; major changes in presentation

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Cited by in corpus (10)

• Multiplicative Convolution of Real Asymmetric and Real Antisymmetric Matrices
• Harmonic analysis for rank-1 Randomised Horn Problems
• Spectral statistics for the difference of two Wishart matrices
• Derivative principles for invariant ensembles
• Hirschman-Widder densities
• Hard Edge Statistics of Products of Pólya Ensembles and Shifted GUE's
• Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance
• Hard edge asymptotics of correlation functions between singular values and eigenvalues
• Correlation functions between singular values and eigenvalues
• Average mutual information for random fermionic Gaussian quantum states