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We investigate the statistical properties of C = uvu−1v−1, when u and v are independent random matrices, uniformly distributed with respect to the Haar measure of the groups U(N) and O(N). An exact formula is derived for the average value of power sum symmetric functions of C, and also for products of the matrix elements of C, similar to Weingarten functions. The density of eigenvalues of C is shown to become constant in the large-N limit, and the first N−1 correction is found. [ABSTRACT FROM AUTHOR]

We consider the problem of electronic quantum transport through ballistic mesoscopic systems with chaotic dynamics, connected to a three-terminal architecture in which one of the terminals has a tunnel barrier. Using a semiclassical approximation based on matrix integrals, we calculate several transport statistics, such as average and variance of conductance, average shot-noise power, among others, that give access to the extreme quantum regime (small channel numbers in the terminal) for broken and intact time-reversal symmetry, which the traditional random matrix approach does not access. As an application, we treat the dephasing regime. [ABSTRACT FROM AUTHOR]

We consider the problem of a semiclassical description of quantum chaotic transport, when a tunnel barrier is present in one of the leads. Using a semiclassical approach formulated in terms of a matrix model, we obtain transport moments as power series in the reflection probability of the barrier, whose coefficients are rational functions of the number of open channels M. Our results are therefore valid in the quantum regime and not only when M ≫ 1. The expressions we arrive at are not identical with the corresponding predictions from random matrix theory, but are in fact much simpler. Both theories agree as far as we can test. [ABSTRACT FROM AUTHOR]

In this article I present an introduction to the problem of quantum chaos, i.e. the characterization and treatment of quantum systems whose classical analogue has chaotic dynamics. Two of the main approaches to this problem are discussed: one is statistical, making use of random matrices, while the other is based on the semiclassical approximation. Results obtained by these approaches are mentioned for closed systems (spectral correlation functions), transport (moments like conductance and shot noise) and resonances (fractal Weyl law and fractal wave functions). [ABSTRACT FROM AUTHOR]

Rotations are among the simplest and most important of linear operations, and the group of rotations is a fundamental structure both for physics and geometry. In the article, we present an introductory discussion about random rotations, which can be seen as an example from the more general theory of random operators or random matrices, area which finds countless applications. We discuss the structure of the orthogonal group. an algorithm for generating random rotations and results for the probability distributions of matrix elements and eigenvalues. [ABSTRACT FROM AUTHOR]

We consider random stochastic matrices M with elements given by Mij = |Uij|2, with U being uniformly distributed on one of the classical compact Lie groups or some of the associated symmetric spaces. We observe numerically that, for large dimensions, the spectral statistics of M, discarding the Perron-Frobenius eigenvalue 1, are similar to those of the Gaussian orthogonal ensemble for symmetric matrices and to those of the real Ginibre ensemble for nonsymmetric matrices. We compute some spectral statistics using Weingarten functions and establish connections with some difficult enumerative problems involving permutations. [ABSTRACT FROM AUTHOR]

We present a semiclassical calculation, based on classical action correlations implemented by means of a matrix integral, of all moments of the Wigner–Smith time delay matrix, Q , in the context of quantum scattering through systems with chaotic dynamics. Our results are valid for broken time reversal symmetry and depend only on the classical dwell time and the number of open channels, M , which is arbitrary. Agreement with corresponding random matrix theory reduces to an identity involving some combinatorial concepts, which can be proved in special cases. • Implements an efficient semiclassical approach in terms of a matrix integral. • Computes general statistics of time delay in quantum chaotic systems. • Advances a new conjecture in the theory of symmetric polynomials. [ABSTRACT FROM AUTHOR]

We consider the statistics of time delay in a chaotic cavity having M open channels, in the absence of time-reversal invariance. In the random matrix theory approach, we compute the average value of polynomial functions of the time delay matrix Q = -iħS†dS/dE, where S is the scattering matrix. Our results do not assume M to be large. In a companion paper, we develop a semiclassical approximation to S-matrix correlation functions, from which the statistics of Q can also be derived. Together, these papers contribute to establishing the conjectured equivalence between the random matrix and the semiclassical approaches. [ABSTRACT FROM AUTHOR]

We propose a matrix model which embodies the semiclassical approach to the problem of quantum transport in chaotic systems. Specifically, a matrix integral is presented whose perturbative expansion satisfies precisely the semiclassical diagrammatic rules for the calculation of general counting statistics. Evaluating it exactly, we show that it agrees with corresponding predictions from random matrix theory. This uncovers the algebraic structure behind the equivalence between these two approaches, and opens the way for further semiclassical calculations. [ABSTRACT FROM AUTHOR]

We review recent studies on the resonance spectrum of quantum scattering systems, in the semiclassical limit and assuming chaotic classical dynamics. Stationary quantum properties are related to fractal structures in the classical phase space. We focus attention on a particular class of problems that are chaotic maps in the focus with holes. Among the topics considered are the fractal Weyl law, the formation of a spectral gap and the morphology of eigenstates. We also discuss the situation where the holes are only partially transparent and the use of random matrices for a statistical description. [ABSTRACT FROM AUTHOR]

Wave scattering in chaotic systems can be characterized by its spectrum of resonances, z&ldquor; = E&ldquor; -- i f, where E&ldquor; is related to the energy and T&ldquor; is the decay rate or width of the resonance. If the corresponding ray dynamics is chaotic, a gap is believed to develop in the large-energy limit: almost all T&ldquor; become larger than some y. However, rare cases with F < y may be present and actually dominate scattering events. We consider the statistical properties of these supersharp resonances. We find that their number does not follow the fractal Weyl law conjectured for the bulk of the spectrum. We also test, for a simple model, the universal predictions of random matrix theory for density of states inside the gap and the hereby derived probability distribution of gap size. [ABSTRACT FROM AUTHOR]

Abstract: We obtain explicit expressions for positive integer moments of the probability density of eigenvalues of the Jacobi and Laguerre random matrix ensembles, in the asymptotic regime of large dimension. These densities are closely related to the Selberg and Selberg-like multidimensional integrals. Our method of solution is combinatorial: it consists in the enumeration of certain classes of lattice paths associated to the solution of recurrence relations. [Copyright &y& Elsevier]