Products of random matrices from polynomial ensembles

Abstract

Very recently we have shown that the spherical transform is a convenient tool for studying the relation between the joint density of the singular values and that of the eigenvalues for bi-unitarily invariant random matrices. In the present work we discuss the implications of these results for products of random matrices. In particular, we derive a transformation formula for the joint densities of a product of two independent bi-unitarily invariant random matrices, the first from a polynomial ensemble and the second from a polynomial ensemble of derivative type. This allows us to re-derive and generalize a number of recent results in random matrix theory, including a transformation formula for the kernels of the corresponding determinantal point processes. Starting from these results, we construct a continuous family of random matrix ensembles interpolating between the products of different numbers of Ginibre matrices and inverse Ginibre matrices. Furthermore, we make contact to the asymptotic distribution of the Lyapunov exponents of the products of a large number of bi-unitarily invariant random matrices of fixed dimension.

Multiplicative convolution of real asymmetric and real anti-symmetric matrices

“…This is no longer true for the real case; see [31] for an extended discussion of the role of group integrals in the computation of the jPDF of the singular values for product matrices of Gaussians or truncated unitaries. Another, closely related, viewpoint has emerged from the recent joint work of Kösters with one of the authors [29,30] which is based on harmonic analysis [23]. Here, generalising the role played by the Mellin transform in the study or products of scalar random variables, a theory of random product matrices based on the spherical transform has been proposed.…”

“…det[cosh x i y j ] i,j=1,...,n ∆ n (x 2 )∆ n (y 2 ) , (1.6) where X, Y are 2n × 2n antisymmetric matrices with singular values {x j }, {y j }, and its analogue for the odd dimensional case. With regard to the harmonic analysis approach [29,30], the natural question of an understanding of explicit formulas in relation to the Hermitised product (1.5) from the viewpoint of spherical transforms arises. The purpose of the present work is to provide such an insight in the case that the matrices are all even dimensional.…”

“…(1.5). For this purpose we introduce our notations and review the main ideas of the harmonic analysis approach of [29,30]. Theorem 2.3 specifies in terms of determinants and Vandermonde products the central object of this study, namely the corresponding spherical function Φ(s; x).…”

From integrable to chaotic systems: Universal local statistics of Lyapunov exponents

“…In several works, e.g. [10,[40][41][42][43], it has been shown for various kinds of X j being i.i.d. that the distributions of the Lyapunov exponents λ j become asymptotically Gaussian when choosing M N .…”

Product Matrix Processes as Limits of Random Plane Partitions

“…It is known that multiplication of (real/complex/quaternion at β = 1, 2, 4) matrices is intimately linked to multiplication of corresponding Jack (=zonal) polynomials, which become Schur polynomials in the case of the complex field (β = 2) that we discuss here. This is discussed by Macdonald [34, Chapter VII], Forrester [20,Section 13.4.3], and more recently used, e.g., by Kieburg, Kosters [29] and by Gorin, Marcus [24]. If we consider a version of the multidimensional Fourier transform for the Schur measures (the appropriate version was introduced by Gorin, Panova [23], Bufetov, Gorin [16] under the name Schur generating functions), then being a Schur measure or its continuous limit is equivalent to the factorization of this transform into a product of one variable functions.…”