Moments of the transmission eigenvalues, proper delay times and random matrix theory I

TL;DR

This paper introduces a method to compute moments of eigenvalue densities in classical random matrix ensembles for all symmetry classes and finite sizes, with applications to quantum transport phenomena.

Contribution

It provides explicit formulas for moments of eigenvalues in Gaussian, Laguerre, and Jacobi ensembles across all symmetry classes beta=1,2,4, including finite matrix size n.

Findings

Derived formulas for eigenvalue moments in classical ensembles.

Connected eigenvalue moments to physical quantities like transmission and delay times.

Formulas facilitate asymptotic analysis as matrix size grows large.

Abstract

We develop a method to compute the moments of the eigenvalue densities of matrices in the Gaussian, Laguerre and Jacobi ensembles for all the symmetry classes beta = 1,2, 4 and finite matrix dimension n. The moments of the Jacobi ensembles have a physical interpretation as the moments of the transmission eigenvalues of an electron through a quantum dot with chaotic dynamics. For the Laguerre ensemble we also evaluate the finite n negative moments. Physically, they correspond to the moments of the proper delay times, which are the eigenvalues of the Wigner-Smith matrix. Our formulae are well suited to an asymptotic analysis as n -> infinity.