Supersymmetry for Products of Random Matrices

TL;DR

This paper develops a novel approach using the projection formula to analyze the singular value statistics of products of random matrices, enabling calculations of local spectral statistics where traditional methods fail.

Contribution

It introduces the projection formula as a shortcut to supersymmetry, allowing the study of local spectral statistics of matrix products, including new results for real and quaternion matrices.

Findings

Calculated the hard edge scaling limit for various random matrix ensembles.

Discovered universality of the hard edge scaling limit for real and quaternion matrix products.

Identified the non-linear σ-models for product 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.