Multiplicative convolution of real asymmetric and real anti-symmetric matrices

Abstract

<jats:title>Abstract</jats:title><jats:p>The singular values of products of standard complex Gaussian random matrices, or sub-blocks of Haar distributed unitary matrices, have the property that their probability distribution has an explicit, structured form referred to as a polynomial ensemble. It is furthermore the case that the corresponding bi-orthogonal system can be determined in terms of Meijer G-functions, and the correlation kernel given as an explicit double contour integral. It has recently been shown that the Hermitised product <jats:inline-formula id="j_apam-2018-0037_ineq_9999_w2aab3b7e1076b1b6b1aab1c14b1b1Aa"><jats:alternatives><jats:tex-math>{X_{M}\cdots X_{2}X_{1}AX_{1}^{T}X_{2}^{T}\cdots X_{M}^{T}}</jats:tex-math></jats:alternatives></jats:inline-formula>, where each <jats:inline-formula id="j_apam-2018-0037_ineq_9998_w2aab3b7e1076b1b6b1aab1c14b1b3Aa"><jats:alternatives><jats:tex-math>{X_{i}}</jats:tex-math></jats:alternatives></jats:inline-formula> is a standard real Gaussian matrix and <jats:italic>A</jats:italic> is real anti-symmetric, exhibits analogous properties. Here we use the theory of spherical functions and transforms to present a theory which, for even dimensions, includes these properties of the latter product as a special case. As an example we show that the theory also allows for a treatment of this class of Hermitised product when the <jats:inline-formula id="j_apam-2018-0037_ineq_9997_w2aab3b7e1076b1b6b1aab1c14b1b7Aa"><jats:alternatives><jats:tex-math>{X_{i}}</jats:tex-math></jats:alternatives></jats:inline-formula> are chosen as sub-blocks of Haar distributed real orthogonal matrices.</jats:p>

On some nonlocal elliptic systems with multiple parameters

“…This theorem is proven in Appendix A.1 in a very similar way as the real counterpart with even dimensional antisymmetric matrices in [23]. This theorem shows that we can deal with all products of the form gxg * with x being m × m Hermitian and g a complex n × m rectangular with m ≤ n in a unified way.…”

“…The theory of spherical transforms [18] developed by Harish-Chandra et al in the 50's has been extremely helpful in dealing with sums [29,16] and products [25,26,23] of random matrices. Regarding products, originally [18] the action of the complex general group G n = Gl C (n) on the cone of positive Hermitian matrices, (g, x) → gxg * has been considered.…”

“…In the present work, we show that there is a natural extension of this setting to the whole space of Hermitian matrices, even those that have not a full rank. However, already when dealing with products of real antisymmetric and real asymmetric matrices, see [23], we have seen that the original definition of the spherical transform does not carry far and needs to be modified. This is also here the case.…”

“…These calculations have been possible due to known group integrals that are involved. Only a few years later the concept of harmonic analysis [18] has been combined with these new developments to enlarge the class of non-classical ensembles to those of Pólya ensembles [16,20,23] (originally known as polynomial ensembles of derivative type [25,26,29]), see Definition 4.1, which also comprise some Muttalib-Borodin ensembles [9,37], for instance.…”

“…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].…”