Commutative law for products of infinitely large isotropic random matrices

Abstract

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 $N\to \infty$. 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 of A^2B^2C^3. We also discuss the singular behavior of the eigenvalue and singular value densities of isotropic matrices and their products for small eigenvalues $\lambda \to 0$. 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 $N\to \infty$: the eigenvalue density of an isotropic random matrix has a power law singularity at the origin $\sim |\lambda {|}^{-s}$ with a power $s\in (0,2)$ when and only when the density of its singular values has a power law singularity $\sim {\lambda }^{-\sigma }$ with a power $\sigma =s/(4-s)$. These results are obtained analytically in the limit $N\to \infty$. 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.

15 pages, 4 figures

References in corpus (4)

• Universal microscopic correlation functions for products of independent Ginibre matrices
• New spectral relations between products and powers of isotropic random matrices
• Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices (The Extended Version)
• On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values

Cited by in corpus (7)

• Weak Commutation Relations and Eigenvalue Statistics for Products of Rectangular Random Matrices
• Relating the Bures measure to the Cauchy two-matrix model
• Universal distribution of Lyapunov exponents for products of Ginibre matrices
• Free products of large random matrices - a short review of recent developments
• Universality of local spectral statistics of products of random matrices
• Eigenvector statistics of the product of Ginibre matrices
• Characteristic polynomials of products of Wigner matrices: finite-N results and Lyapunov universality