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

Abstract

We develop a method to compute the moments of the eigenvalue densities of matrices in the Gaussian, Laguerre and Jacobi ensembles for all the symmetry classes beta = 1,2, 4 and finite matrix dimension n. The moments of the Jacobi ensembles have a physical interpretation as the moments of the transmission eigenvalues of an electron through a quantum dot with chaotic dynamics. For the Laguerre ensemble we also evaluate the finite n negative moments. Physically, they correspond to the moments of the proper delay times, which are the eigenvalues of the Wigner-Smith matrix. Our formulae are well suited to an asymptotic analysis as n -> infinity.

33 pages, typos corrected

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Cited by in corpus (39)

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• Moments of the Gaussian $\beta$ Ensembles and the large-$N$ expansion of the densities
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• Universality in chaotic quantum transport: The concordance between random matrix and semiclassical theories
• Efficient semiclassical approach for time delays
• Transport moments beyond the leading order
• Moments of random matrices and hypergeometric orthogonal polynomials
• Combinatorial theory of the semiclassical evaluation of transport moments I: Equivalence with the random matrix approach
• Correlators for the Wigner-Smith time-delay matrix of chaotic cavities
• Statistical distribution of the Wigner-Smith time-delay matrix moments for 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
• Time-delay matrix, midgap spectral peak, and thermopower of an Andreev billiard
• 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
• Linear Differential Equations for the Resolvents of the Classical Matrix Ensembles
• Joint statistics of quantum transport in chaotic cavities
• Semiclassical treatment of quantum chaotic transport with a tunnel barrier
• Relations between moments for the Jacobi and Cauchy random matrix ensembles
• Entanglement production in non-ideal cavities and optimal opacity
• The Correlated Jacobi and the Correlated Cauchy-Lorentz ensembles
• Random density matrices: Analytical results for mean root fidelity and mean square Bures distance
• Statistics of quantum transport in weakly non-ideal chaotic cavities
• Time delay statistics for finite number of channels in all symmetry classes
• Wigner-Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity
• Random density matrices: Analytical results for mean fidelity and variance of squared Bures distance
• 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
• Delay times in chaotic quantum systems
• Integrable Aspects of Universal Quantum Transport in Chaotic Cavities