Weak commutation relations and eigenvalue statistics for products of rectangular random matrices

Abstract

We study the joint probability density of the eigenvalues of a product of rectangular real, complex or quaternion random matrices in a unified way. The random matrices are distributed according to arbitrary probability densities, whose only restriction is the invariance under left and right multiplication by orthogonal, unitary or unitary symplectic matrices, respectively. We show that a product of rectangular matrices is statistically equivalent to a product of square matrices. Hereby we prove a weak commutation relation of the random matrices at finite matrix sizes, which previously have been discussed for infinite matrix size. Moreover we derive the joint probability densities of the eigenvalues. To illustrate our results we apply them to a product of random matrices drawn from Ginibre ensembles and Jacobi ensembles as well as a mixed version thereof. For these weights we show that the product of complex random matrices yield a determinantal point process, while the real and quaternion matrix ensembles correspond to Pfaffian point processes. Our results are visualized by numerical simulations. Furthermore, we present an application to a transport on a closed, disordered chain coupled to a particle bath.

Supersymmetry for Products of Random Matrices

“…(13) we have chosen a supersymmetric extension of P to the superspace gl (β) (n + γl|γl; n|0) which is by far not unique. However the final result is independent of this choice as already discussed in [12]. Such an extension indeed exists for a smooth distribution P .…”

“…We assume ν = 0 in the following to keep the computations simple such that we choose the abbreviation gl (β) (n) = gl (β) (n; n). Nonetheless this restriction is not that strong since a product of rectangular matrices can be always rephrased to a product of square matrices [12]. Examples of such induced measures resulting from rectangular matrices are given in Sec.…”

“…Therefore one can understand Eq. (24) as an induced measure [12]. The corresponding weight Q WL is given by Eq.…”

“…For example, free probability has proven as an efficient tool for calculating the macroscopic level density [9]. With the help of orthogonal polynomials one could calculate algebraic structures like determinants and Pfaffians, their kernels, and certain universal statistics on the local scale of the spectrum [6,10,11,12,13]. In particular products of random matrices drawn from Meijer G-ensembles (the weight is essentially given by Meijer G-functions, see [14] for a definition of these functions) exhibit a new kind of universal kernel in the hard scaling limit (microscopic limit around the origin).…”

The Correlated Jacobi and the Correlated Cauchy–Lorentz Ensembles

“…This identity is reminiscent of the weak-commutation relation proven in [36]. When rescaling B → B F † the integral over F becomes a deformed Gaussian and reads…”