Matrix Product Ensembles of Hermite Type and the Hyperbolic Harish-Chandra–Itzykson–Zuber Integral

Abstract

We investigate spectral properties of a Hermitised random matrix product which, contrary to previous product ensembles, allows for eigenvalues on the full real line. We prove that the eigenvalues form a bi-orthogonal ensemble, which reduces asymptotically to the Hermite Muttalib-Borodin ensemble. Explicit expressions for the bi-orthogonal functions as well as the correlation kernel are provided. Scaling the latter near the origin gives a limiting kernel involving Meijer G-functions, and the functional form of the global density is calculated. As a part of this study, we introduce a new matrix transformation which maps the space of polynomial ensembles onto itself. This matrix transformation is closely related to the so-called hyperbolic Harish-Chandra-Itzykson-Zuber integral. *

On some nonlocal elliptic systems with multiple parameters

“…They slightly differ from the original definition by Harish-Chandra et al, see [18] and references therein, in the normalization and the duplication of the complex plane of the "Mellin-Fourier" (frequency) parameter s. In total, one needs 2 r copies for a matrix of rank r denoted by the vector L ∈ Z r 2 {0, 1} r . These copies are essential to keep the information of the number of positive and negative eigenvalues, since this number stays fixed in such a product as already observed in [14]. In spite of these two modifications of the spherical transform, the results resemble very much the original results for matrices on the general (special) linear group [18,25].…”

“…This is the case, for instance, for the Ginibre ensemble where Mω(s) = Γ(s) and for the Jacobi ensemble with Mω(s) = Γ(s + ν)Γ(μ + n)/Γ(s + ν + μ + n) for ν > −1 and μ > 0, see [27,26]. The case of the product of an induced Ginibre ensemble whose weight is equal to ω(a) = x ν e −a has been studied in [14]; in particular Theorem 4.7 and Proposition 4.8, up to normalization, become Lemma 2 and Proposition 7 in [14], respectively. The statements above generalize this discussion and avoids the non-compact group integrals encountered in [14] that are unknown for more general weights, e.g., for the Jacobi ensemble with ω(a) = a ν (1 − a) μ+n− 1 Θ(1 − a), Cauchy-Lorentz ensemble with ω(a) = a ν / (1 + a) μ+n or the Muttalib-Borodin ensembles like ω(a) = a ν e −a θ with θ > 0 or ω(a) = a ν e −(ln a) 2 .…”

“…The case of the product of an induced Ginibre ensemble whose weight is equal to ω(a) = x ν e −a has been studied in [14]; in particular Theorem 4.7 and Proposition 4.8, up to normalization, become Lemma 2 and Proposition 7 in [14], respectively. The statements above generalize this discussion and avoids the non-compact group integrals encountered in [14] that are unknown for more general weights, e.g., for the Jacobi ensemble with ω(a) = a ν (1 − a) μ+n− 1 Θ(1 − a), Cauchy-Lorentz ensemble with ω(a) = a ν / (1 + a) μ+n or the Muttalib-Borodin ensembles like ω(a) = a ν e −a θ with θ > 0 or ω(a) = a ν e −(ln a) 2 . All of these examples are Pólya ensembles, and our theoretical framework and its results deals with them in a unifying way.…”

“…Very recently, products of real square matrices with real antisymmetric matrices and of quaternion matrices with anti-selfdual anti-Hermitian matrices have been studied [15,23]. These works have followed an article on products of complex Ginibre matrices with a Gaussian unitary ensemble (GUE) [14]. Therein it has been shown that the natural group action of the matrix group G on the corresponding Liealgebra of its maximal compact subgroup might be indeed analytically feasible.…”

Spectral statistics for the difference of two Wishart matrices

“…It is clear from equation (22) that w(λ) is analytic for both λ < 0 and λ > 0, and the derivatives in these regions can be obtained readily. The expressions given in (33) and (35) follow from (24) and (26) by applying Rodrigues formula [86],…”