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

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-Büttiker scattering matrix for chaotic ballistic cavities can be modelled by the circular ensembles of random matrix theory. The starting points are the finite-n formulae that we recently discovered [F. Mezzadri and N. J. Simm, “Moments of the transmission eigenvalues, proper delay times and random matrix theory,” J. Math. Phys. 52, 103511 (2011)]10.1063/1.3644378 . Our analysis includes all the symmetry classes β ∈ {1, 2, 4}; in addition, it applies to the transmission eigenvalues of Andreev billiards, whose symmetry classes were classified by Zirnbauer [“Riemannian symmetric superspaces and their origin in random-matrix theory,” J. Math. Phys. 37(10), 4986 (1996)]10.1063/1.531675 and Altland and Zirnbauer [“Random matrix theory of a chaotic Andreev quantum dot,” Phys. Rev. Lett. 76(18), 3420 (1996)10.1103/PhysRevLett.76.3420 ; Altland and Zirnbauer “Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures,” Phys. Rev. B 55(2), 1142 (1997)]10.1103/PhysRevB.55.1142 . Where applicable, our results are in complete agreement with the semiclassical theory of mesoscopic systems developed by Berkolaiko et al. [“Full counting statistics of chaotic cavities from classical action correlations,” J. Phys. A: Math. Theor. 41(36), 365102 (2008)]10.1088/1751-8113/41/36/365102 and Berkolaiko and Kuipers [“Moments of the Wigner delay times,” J. Phys. A: Math. Theor. 43(3), 035101 (2010)10.1088/1751-8113/43/3/035101 ; Berkolaiko and Kuipers “Transport moments beyond the leading order,” New J. Phys. 13(6), 063020 (2011)]10.1088/1367-2630/13/6/063020 . Our approach also applies to the Selberg-like integrals. We calculate the first two terms in their asymptotic expansion explicitly.