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

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-Bütticker 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.

45 pages; typos corrected and 6 references added

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Cited by in corpus (32)

• Wigner time delay and related concepts -- Application to transport in coherent conductors
• Tau-Function Theory of Quantum Chaotic Transport with beta=1,2,4
• Efficient semiclassical approach for time delays
• Moments of random matrices and hypergeometric orthogonal polynomials
• Combinatorial theory of the semiclassical evaluation of transport moments I: Equivalence with the random matrix approach
• Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities
• Correlators for the Wigner-Smith time-delay matrix of chaotic cavities
• Large $N$ expansions for the Laguerre and Jacobi $\beta$ ensembles from the loop equations
• Moments of the eigenvalue densities and of the secular coefficients of $\beta$-ensembles
• Large-$N$ expansion for the time-delay matrix of ballistic chaotic cavities
• Recursion for the smallest eigenvalue density of $\beta$-Wishart-Laguerre ensemble
• Integer moments of complex Wishart matrices and Hurwitz numbers
• Wigner-Smith time-delay matrix in chaotic cavities with non-ideal contacts
• Large deviation eigenvalue density for the soft edge Laguerre and Jacobi $\beta$-ensembles
• Transport moments and Andreev billiards with tunnel barriers
• Combinatorial theory of the semiclassical evaluation of transport moments II: Algorithmic approach for moment generating functions
• Statistics of time delay and scattering correlation functions in chaotic systems I. Random Matrix Theory
• Joint moments of proper delay times
• Joint statistics of quantum transport in chaotic cavities
• Linear Differential Equations for the Resolvents of the Classical Matrix Ensembles
• Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials
• Entanglement production in non-ideal cavities and optimal opacity
• The Correlated Jacobi and the Correlated Cauchy-Lorentz ensembles
• Time delay statistics for finite number of channels in all symmetry classes
• Statistics of quantum transport in weakly non-ideal chaotic cavities
• Wigner-Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity
• Semiclassical approach to $S$ matrix energy correlations and time delay in chaotic systems
• Spectral statistics of the uni-modular ensemble
• The probability distribution of spectral moments for the Gaussian beta-ensembles
• Semiclassical calculation of time delay statistics in chaotic quantum scattering
• Integrable Aspects of Universal Quantum Transport in Chaotic Cavities
• Delay times in chaotic quantum systems