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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]

When time-reversal symmetry is broken, the average conductance through a chaotic cavity, from an entrance lead with N 1 open channels to an exit lead with N 2 open channels, is given by N 1 N 2/ M, where M = N 1 + N 2. We show that, when tunnel barriers of reflectivity Îł are placed on the leads, two correction terms appear in the average conductance, and that one of them is proportional to Îł M . Since M ∼ â„Ź âˆ'1, this correction is exponentially small in the semiclassical limit. We derive this term from a semiclassical approximation, generally expected to give only leading orders in powers of â„Ź. Even though the theory is built perturbatively both in Îł and in 1/ M, the final result is exact. [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]

Within a random matrix theory approach, we obtain spectral statistics of the Wigner time delay matrix Q, for arbitrary channels number M and for all symmetry classes, in fact for the general Dyson parameter β. We also put forth two conjectures: one is related to the large- M expansion of joint cumulants of traces of powers of Q, which generalizes and implies a previous conjecture of Cunden, Mezzadri, Vivo and Simm; the other concerns the tail of the distribution of traces of powers of Q. [ABSTRACT FROM AUTHOR]

We present a pedagogical treatment of probability distributions, which is central for the understanding of statistical mechanics and quantum mechanics, apart from being importante for every kind of statistical description of natural phenomena. [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]

Arthur Eddington, one of the main proponents and principal manager of the british expeditions that observed the 1919 eclipse in Sobral and in the Island of Principe, put down his observations and the results obtained in the photographic plates in his classical book Space, Time and Gravitation. This article consists in translation and notes on the fundamental participation of Eddington in validating the gravitational deflection of light, one of the predictions from Einstein's theory of general relativity. [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]

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]

A semiclassical approach to the calculation of transport moments Mm = Tr[(t† t )m], where t is the transmission matrix, was developed by this author (2012 Europhys. Lett. 98 20006) for chaotic cavities with two leads and broken time-reversal symmetry. The result is an expression for Mm as a perturbation series in 1/N, where N is the total number of open channels, which is in agreement with random matrix theory predictions. The coefficients in this series were related to two open combinatorial problems. Here we expand on this work, including the solution to one of the combinatorial problems. As a by-product, we also present a conjecture relating two kinds of factorizations of permutations. [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]

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]

Starting from a semiclassical approach recently developed for spectral correlation functions of quantum systems whose classical dynamics is chaotic, we focus on the case of broken time-reversal symmetry, the so-called unitary class. We obtain to all orders in perturbation theory the nonoscillatory parts of all correlation functions, showing that the off-diagonal contributions to these correlation functions cancel and the conjectured universality holds. The innovation that allows this calculation to be performed is the introduction of an auxiliary matrix model which is governed by the same diagrammatic rules as the semiclassical approach and which can be exactly solved. [ABSTRACT FROM AUTHOR]

We present a semiclassical approach to n-point spectral correlation functions of quantum systems whose classical dynamics is chaotic, for arbitrary n. The basic ingredients are sets of periodic orbits that have nearly the same action and therefore provide constructive interference. We calculate explicitly the first correlation functions, to leading orders in their energy arguments, for both unitary and orthogonal symmetry classes. The results agree with corresponding predictions from random matrix theory, thereby giving solid support to the conjecture of universality. [ABSTRACT FROM AUTHOR]

We investigate entanglement production in a generic normal conductor, connected to two single-channel left leads and two single-channel right leads. We consider the joint statistics of the number of entangled pairs produced during a given time and amount of entanglement carried by them. We find a constraint that implies that more entangled states are less likely to be detected. Namely, if C is the concurrence and N is the squared norm of the entangled state, then production occurs only if N(1+C)<1. For the particular case of a chaotic cavity working as quantum entangler, we obtain explicit expressions for the joint distribution of C and N, both for systems with and without time-reversal symmetry. [ABSTRACT FROM AUTHOR]

The statistics of quantum transport through chaotic cavities with two leads is encoded in transport moments Mm=Tr[(t†t)m], where t is the transmission matrix, which have a known universal expression for systems without time-reversal symmetry. We present a semiclassical derivation of this universality, based on action correlations that exist between sets of long scattering trajectories. Our semiclassical formula for Mm holds for all values of m and an arbitrary number of open channels. This is achieved by mapping the problem into two independent combinatorial problems, one involving pairs of set partitions and the other involving factorizations in the symmetric group. [ABSTRACT FROM AUTHOR]

We consider statistics of electronic transport in chaotic cavities where time-reversal symmetry is broken and one of the leads is weakly nonideal; that is, it contains tunnel barriers characterized by tunneling probabilities Γ i . Using symmetric function expansions and a generalized Selberg integral, we develop a systematic perturbation theory in 1-Γ i valid for an arbitrary number of channels and obtain explicit formulas up to second order for the average and variance of the conductance and for the average shot noise. Higher moments of the conductance are considered to leading order. [ABSTRACT FROM AUTHOR]