Supersymmetry for Products of Random Matrices

Abstract

We consider the singular value statistics of products of independent random matrices. In particular we compute the corresponding averages of products of characteristic polynomials. To this aim we apply the projection formula recently introduced for chiral random matrix ensembles which serves as a short cut of the supersymmetry method. The projection formula enables us to study the local statistics where free probability currently fails. To illustrate the projection formula and underlining the power of our approach we calculate the hard edge scaling limit of the Meijer G-ensembles comprising the Wishart-Laguerre (chiral Gaussian), the Jacobi (truncated orthogonal, unitary or unitray symplectic) and the Cauchy-Lorentz (heavy tail) random matrix ensembles. All calculations are done for real, complex, and quaternion matrices in a unifying way. In the case of real and quaternion matrices the results are completely new and point to the universality of the hard edge scaling limit for a product of these matrices, too. Moreover we identify the non-linear $\sigma$-models corresponding to product matrices.

content partially presented in a talk at "Random Matrix Theory: Foundations and Applications" in Cracow, July 1-6 2014,18 pages, PACS: 02.10.Yn, 02.50.Sk, 05.40.-a

References in corpus (2)

• Recent exact and asymptotic results for products of independent random matrices
• The supersymmetry method of random matrix theory

Cited by in corpus (6)

• Exact Relation between Singular Value and Eigenvalue Statistics
• Products of Independent Gaussian Random Matrices
• Spectral correlation functions of the sum of two independent complex Wishart matrices with unequal covariances
• The Correlated Jacobi and the Correlated Cauchy-Lorentz ensembles
• Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance
• Characteristic polynomials of products of Wigner matrices: finite-N results and Lyapunov universality