Commutative law for products of infinitely large isotropic random matrices

Ensembles of isotropic random matrices are defined by the invariance of the probability measure under the left (and right) multiplication by an arbitrary unitary matrix. We show that the multiplication of large isotropic random matrices is spectrally commutative and self-averaging in the limit of infinite matrix size N→∞. The notion of spectral commutativity means that the eigenvalue density of a product ABC... of such matrices is independent of the order of matrix multiplication, for example, the matrix ABCD has the same eigenvalue density as ADCB. In turn, the notion of self-averaging means that the product of n independent but identically distributed random matrices, which we symbolically denote by AAA..., has the same eigenvalue density as the corresponding power A(n) of a single matrix drawn from the underlying matrix ensemble. For example, the eigenvalue density of ABCCABC is the same as that of A(2)B(2)C(3). We also discuss the singular behavior of the eigenvalue and singular value densities of isotropic matrices and their products for small eigenvalues λ→0. We show that the singularities at the origin of the eigenvalue density and of the singular value density are in one-to-one correspondence in the limit N→∞: The eigenvalue density of an isotropic random matrix has a power-law singularity at the origin ~|λ|(-s) with a power sε(0,2) when and only when the density of its singular values has a power-law singularity ~λ(-σ) with a power σ=s/(4-s). These results are obtained analytically in the limit N→∞. We supplement these results with numerical simulations for large but finite N and discuss finite-size effects for the most common ensembles of isotropic random matrices.