1205.1625 (Z. Burda et al.)

New spectral relations between products and powers of isotropic random
matrices    [PDF]

Z. Burda, M. A. Nowak, A. Swiech

We show that the limiting eigenvalue density of the product of $n$ identically distributed random matrices from an isotropic unitary ensemble (IUE) is equal to the eigenvalue density of n-th power of a single matrix from this ensemble, in the limit when the size of the matrix tends to infinity. Using this observation one can derive the limiting density of the product of n independent identically distributed matrices for many physically interesting matrix ensembles with invariant measures given by the partition function of the form $Z = \int Dx \exp - N {\rm Tr} V(x^\dagger x)$. In this paper we discuss two examples: the product of $n$ Girko-Ginibre matrices and the product of n truncated unitary matrices.

View original: http://arxiv.org/abs/1205.1625