Transmission eigenvalue densities and moments in chaotic cavities from random matrix theory

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 λ m 1 for m > −|N 1 − N 2 | − 1 and β = 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 β = 2, 4 which does not make use of the orthogonal polynomials technique. This result may be relevant for an efficient numerical implementation avoiding determinants.

Statistics of quantum transport in chaotic cavities with broken time-reversal symmetry

“…It has the merit of having only n terms, in contrast with the result in Ref. 13 where the number of terms in the sum grows with N 1 and N 2 . In Fig.…”

“…8 More recently, results were found for: the linear statistics ͗T 2 ͘ and ͗T 3 ͘, 9,11 some nonlinear statistics such as variance of shot-noise, skewness, and kurtosis of conductance, 12 the density of eigenvalues for unequal leads, 13 as well as an expression for ͗T n ͘. The purpose of this paper is to establish general exact results, of which several of the abovementioned ones are particular cases, for systems without time-reversal ͑TR͒ symmetry.…”

“…By integrating out all but one of the variables, one obtains the eigenvalue density ͑T͒. An exact expression for ͑T͒ can be written down using Jacobi polynomials, 13,15 …”

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“…Using these results, we have computed the average of integer moments, reducing the complexity from a N -fold integration to a single integral over the spectral density. In all cases, expressions different from ours and derived through different methods already exist [24,31,32] (see also Appendix A). It would be interesting to prove mathematically the equiv-alence of various formulae which are now available for the same objects.…”

Combinatorial problems in the semiclassical approach to quantum chaotic transport

“…Initially, only perturbative results were obtained [9,10], valid to leading or next-toleading order in 1/N , and only for the first moments. In recent years the connection with the Selberg integral was properly realized [11] and this eventually allowed the calculation of all M m to be carried out for arbitrary values of N 1 and N 2 , for both symmetry classes [12,13] (see also [14]). …”