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

TL;DR

This paper analyzes the asymptotic behavior of moments of transmission eigenvalues and proper delay times in quantum chaotic systems using Random Matrix Theory, extending results across symmetry classes and connecting to semiclassical and Selberg integral theories.

Contribution

It provides the first three terms of asymptotic expansions for these moments across all symmetry classes, including applications to Andreev billiards and Selberg-like integrals.

Findings

Asymptotic expansions match semiclassical predictions.

Results apply to all symmetry classes beta=1,2,4.

Explicit calculations for the first two terms of Selberg-like integrals.

Abstract

We systematically study the first three terms in the asymptotic expansions of the moments of the transmission eigenvalues and proper delay times as the number of quantum channels n in the leads goes to infinity. The computations are based on the assumption that the Landauer-B\"utticker scattering matrix for chaotic ballistic cavities can be modelled by the circular ensembles of Random Matrix Theory (RMT). The starting points are the finite-n formulae that we recently discovered (Mezzadri and Simm, J. Math. Phys. 52 (2011), 103511). Our analysis includes all the symmetry classes beta=1,2,4; in addition, it applies to the transmission eigenvalues of Andreev billiards, whose symmetry classes were classified by Zirnbauer (J. Math. Phys. 37 (1996), 4986-5018) and Altland and Zirnbauer (Phys. Rev. B. 55 (1997), 1142-1161). Where applicable, our results are in complete agreement with the semiclassical theory of mesoscopic systems developed by Berkolaiko et al. (J. Phys. A.: Math. Theor. 41 (2008), 365102) and Berkolaiko and Kuipers (J. Phys. A: Math. Theor. 43 (2010), 035101 and New J. Phys. 13 (2011), 063020). Our approach also applies to the Selberg-like integrals. We calculate the first two terms in their asymptotic expansion explicitly.