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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 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]

The M-dimensional unitary matrix S(E), which describes scattering of waves, is a strongly fluctuating function of the energy for complex systems such as ballistic cavities, whose geometry induces chaotic ray dynamics. Its statistical behaviour can be expressed by means of correlation functions of the kind 〈Sij(E + ε)Spq†(E - ε)〉, which have been much studied within the random matrix approach. In this work, we consider correlations involving an arbitrary number of matrix elements and express them as infinite series in 1/M, whose coefficients are rational functions of ε. From a mathematical point of view, this may be seen as a generalization of the Weingarten functions of circular ensembles. [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 consider S-matrix correlation functions for a chaotic cavity having M open channels, in the absence of time-reversal invariance. Relying on a semiclassical approximation, we compute the average over E of the quantities Tr[S†(E - ε)S(E + ε)]n, for general positive integer n. Our result is an infinite series in ., whose coefficients are rational functions of M. From this, we extract moments of the time delay matrix Q = -iħS†dS/dE and check that the first 8 of them agree with the random matrix theory prediction from our previous paper [M. Novaes, J. Math. Phys. 56, 062110 (2015)]. [ABSTRACT FROM AUTHOR]

The semiclassical approximation to the coherent state propagator requires complex classical trajectories in order to satisfy the associated boundary conditions, but finding these trajectories in practice is a difficult task that may compromise the applicability of the approximation. In this work several approximations to the coherent state propagator are derived that make use only of real trajectories, which are easier to handle and have a more direct physical interpretation. It is verified in a particular example that these real trajectories approximations may have excellent accuracy. [ABSTRACT FROM AUTHOR]