Transmission Eigenvalue Densities and Moments in Chaotic Cavities from Random Matrix Theory

TL;DR

This paper derives exact formulas for transmission eigenvalue densities and moments in chaotic cavities using Random Matrix Theory, providing improved and alternative methods for calculating these quantities.

Contribution

It offers a non-perturbative expression for moments of transmission eigenvalues and an independent derivation of correlation functions, enhancing analytical tools in the field.

Findings

Exact moments for eigenvalues with m > -|N1-N2|-1 and β=2 derived.

Analytical eigenvalue densities and correlations obtained from Jacobi ensemble.

Alternative derivation methods that avoid orthogonal polynomials and determinants.

Abstract

We point out that the transmission eigenvalue density and higher order correlation functions in chaotic cavities for an arbitrary number of incoming and outgoing leads $(N_1,N_2)$ are analytically known from the Jacobi ensemble of Random Matrix Theory. Using this result and a simple linear statistic, we give an exact and non-perturbative expression for moments of the form $<\lambda_1^m>$ for $m>-|N_1-N_2|-1$ and $\beta=2$, thus improving the existing results in the literature. Secondly, we offer an independent derivation of the average density and higher order correlation functions for $\beta=2,4$ which does not make use of the orthogonal polynomials technique. This result may be relevant for an efficient numerical implementation avoiding determinants.