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ABSTRACT: This is a concise review of the complex, real and quaternion real Ginibre random matrix ensembles and their elliptic deformations. Eigenvalue correlations are exactly reduced to two-point kernels and discussed in the strongly and weakly non-Hermitian limits of large matrix size. Comment: 23 pages, invited article for the Oxford Handbook of Random Matrix Theory
11/2009;
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ABSTRACT: Ensembles of random density matrices determined by various probability measures are analysed. A simple and efficient algorithm to generate at random density matrices distributed according to the Bures measure is proposed. This procedure may serve as an initial step in performing Bayesian approach to quantum state estimation based on the Bures prior. We study the distribution of purity of random mixed states. The moments of the distribution of purity are determined for quantum states generated with respect to the Bures measure. This calculation serves as an exemplary application of the "deform-and-study" approach based on ideas of integrability theory. It is shown that Painlevé equation appeared as a part of the presented theory.
10/2009;
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ABSTRACT: Applying random matrix theory to quantum transport in chaotic cavities, we develop a novel approach to computation of the moments of the conductance and shot-noise (including their joint moments) of arbitrary order and at any number of open channels. The method is based on the Selberg integral theory combined with the theory of symmetric functions and is applicable equally well for systems with and without time-reversal symmetry. We also compute higher-order cumulants and perform their detailed analysis. In particular, we establish an explicit form of the leading asymptotic of the cumulants in the limit of the large channel numbers. We derive further a general Pfaffian representation for the corresponding distribution functions. The Edgeworth expansion based on the first four cumulants is found to reproduce fairly accurately the distribution functions in the bulk even for a small number of channels. As the latter increases, the distributions become Gaussian-like in the bulk but are always characterized by a power-law dependence near their edges of support. Such asymptotics are determined exactly up to linear order in distances from the edges, including the corresponding constants. Comment: 14 pages, 4 figures, 3 tables
05/2009;
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ABSTRACT: We report on an analytical study of the statistics of conductance, $g$, and shot-noise power, $p$, for a chaotic cavity with arbitrary numbers $N_{1,2}$ of channels in two leads and symmetry parameter $\beta = 1,2,4$. With the theory of Selberg's integral the first four cumulants of $g$ and first two cumulants of $p$ are calculated explicitly. We give analytical expressions for the conductance and shot-noise distributions and determine their exact asymptotics near the edges up to linear order in distances from the edges. For $0<g<1$ a power law for the conductance distribution is exact. All results are also consistent with numerical simulations.
11/2007;
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ABSTRACT: Ensembles of random stochastic and bistochastic matrices are investigated. While all columns of a random stochastic matrix can be chosen independently, the rows and columns of a bistochastic matrix have to be correlated. We evaluate the probability measure induced into the Birkhoff polytope of bistochastic matrices by applying the Sinkhorn algorithm to a given ensemble of random stochastic matrices. For matrices of order N=2 we derive explicit formulae for the probability distributions induced by random stochastic matrices with columns distributed according to the Dirichlet distribution. For arbitrary $N$ we construct an initial ensemble of stochastic matrices which allows one to generate random bistochastic matrices according to a distribution locally flat at the center of the Birkhoff polytope. The value of the probability density at this point enables us to obtain an estimation of the volume of the Birkhoff polytope, consistent with recent asymptotic results. Comment: 22 pages, 4 figures
11/2007;
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ABSTRACT: In the framework of the random matrix approach, we apply the theory of Selberg's integral to problems of quantum transport in chaotic cavities. All the moments of transmission eigenvalues are calculated analytically up to the fourth order. As a result, we derive exact explicit expressions for the skewness and kurtosis of the conductance and transmitted charge as well as for the variance of the shot-noise power in chaotic cavities. The obtained results are generally valid at arbitrary numbers of propagating channels in the two attached leads. In the particular limit of large (and equal) channel numbers, the shot-noise variance attends the universal value 1/(64\beta) that determines a universal Gaussian statistics of shot-noise fluctuations in this case. Comment: 4 pages, 1 figure
11/2007;
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ABSTRACT: The density of stationary points and minima of a N ≫ 1 dimensional Gaussian energy landscape has been calculated. It is used to show that the point of zero-temperature replica
symmetry breaking in the equilibrium statistical mechanics of a particle placed in such a landscape in a spherical box of
size L = R √N corresponds to the onset of exponential in N growth of the cumulative number of stationary points, but not necessarily the minima. For finite temperatures, a simple variational
upper bound on the true free energy of the R = ∞ version of the problem has been constructed and it has been shown that this approximation can recover the position of
the whole de-Almeida-Thouless line.
JETP Letters 04/2007; 85(5):261-266. · 1.35 Impact Factor
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ABSTRACT: We calculate the density of stationary points and minima of a $N\gg 1$ dimensional Gaussian energy landscape. We use it to show that the point of zero-temperature replica symmetry breaking in the equilibrium statistical mechanics of a particle placed in such a landscape in a spherical box of size $L=R\sqrt{N}$ corresponds to the onset of exponential in $N$ growth of the cumulative number of stationary points, but not necessarily the minima. For finite temperatures we construct a simple variational upper bound on the true free energy of the $R=\infty$ version of the problem and show that this approximation is able to recover the position of the whole de-Almeida-Thouless line. Comment: a revised and shortened version with a few typos corrected and references added. To appear in JETP Letters
11/2006;
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ABSTRACT: Using the random matrix approach, we calculate analytically the average shot-noise power in a chaotic cavity at an arbitrary number of propagating modes (channels) in each of the two attached leads. A simple relationship between this quantity, the average conductance and the conductance variance is found. The dependence of the Fano factor on the channel number is considered in detail.
01/2006;
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ABSTRACT: We review recent progress in analysing wave scattering in systems with both intrinsic chaos and/or disorder and internal losses, when the scattering matrix is no longer unitary. By mapping the problem onto a nonlinear supersymmetric sigma-model, we are able to derive closed form analytic expressions for the distribution of reflection probability in a generic disordered system. One of the most important properties resulting from such an analysis is statistical independence between the phase and the modulus of the reflection amplitude in every perfectly open channel. The developed theory has far-reaching consequences for many quantities of interest, including local Green functions and time delays. In particular, we point out the role played by absorption as a sensitive indicator of mechanisms behind the Anderson localisation transition. We also provide a random-matrix-based analysis of S-matrix and impedance correlations for various symmetry classes as well as the distribution of transmitted power for systems with broken time-reversal invariance, completing previous works on the subject. The results can be applied, in particular, to the experimentally accessible impedance and reflection in a microwave or ultrasonic cavity attached to a system of antennas.
08/2005;
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ABSTRACT: Wave scattering in chaotic systems with a uniform energy loss (absorption) is considered. Within the random matrix approach we calculate exactly the energy correlation functions of different matrix elements of impedance or scattering matrices for systems with preserved or broken time-reversal symmetry. The obtained results are valid at any number of arbitrary open scattering channels and arbitrary absorption. Elastic enhancement factors (defined through the ratio of the corresponding variance in reflection to that in transmission) are also discussed.
07/2005;
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ABSTRACT: We establish a general relation between the statistics of the local Green's function for systems with chaotic wave scattering and a uniform energy loss (absorption) and its two-point correlation function for the same system without absorption. Within the random matrix approach this kind of a fluctuation dissipation relation allows us to derive the explicit analytical expression for the joint distribution function of the real and imaginary parts of the local Green function for all symmetry classes as well as at an arbitrary degree of the time-reversal symmetry breaking in the system. The outstanding problem of the orthogonal symmetry is further reduced to simple quadratures. The results can be applied, in particular, to the experimentally accessible impedance and reflection in a microwave cavity attached to a single-mode antenna.
03/2005;
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ABSTRACT: We establish a general relation between the statistics of the local Green’s function for systems with chaotic wave scattering
and uniform energy loss (absorption) and the two-point correlator of its resolvents for the same system without absorption.
Within the random matrix approach, this kind of a fluctuation dissipation relation allows us to derive the explicit analytic
expression for the joint distribution function of the real and imaginary part of the local Green’s function for all symmetry
classes as well as at an arbitrary degree of time-reversal symmetry breaking in the system. The outstanding problem of orthogonal
symmetry is further reduced to simple quadratures. The results can be applied, in particular, to the experimentally accessible
impedance and reflection in a microwave cavity attached to a single-mode antenna.
JETP Letters 01/2005; 82(8):544-548. · 1.35 Impact Factor
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ABSTRACT: The paper discusses progress in understanding statistical properties of complex eigenvalues (and corresponding eigenvectors) of weakly non-unitary and non-Hermitian random matrices. Ensembles of this type emerge in various physical contexts, most importantly in random matrix description of quantum chaotic scattering as well as in the context of QCD-inspired random matrix models. Comment: Published version, with a few more misprints corrected
07/2002;
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ABSTRACT: An ensemble of random unistochastic (orthostochastic) matrices is defined by taking squared moduli of elements of random unitary (orthogonal) matrices distributed according to the Haar measure on U(N) (or O(N), respectively). An ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble. We study the distribution of complex eigenvalues of bistochastic, unistochastic and ortostochastic matrices in the complex plane. We compute averages (entropy, traces) over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices. Comment: 22 latex pages, including 7 figures in ps, minor revisions
12/2001;
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ABSTRACT: The probability distribution of the proper delay times during scattering on a chaotic system is derived in the framework of the random matrix approach and the supersymmetry method. The result obtained is valid for an arbitrary number of scattering channels as well as arbitrary coupling to the energy continuum. The case of statistically equivalent channels is studied in detail. In particular, the semiclassical limit of an infinite number of weak channels is paid appreciable attention.
Physical Review Letters 09/2001; 87(9):094101. · 7.37 Impact Factor
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ABSTRACT: We write explicitly a transformation of the scattering phases reducing the problem of quantum chaotic scattering for systems with M statistically equivalent channels at nonideal coupling to that for ideal coupling. Unfolding the phases by their local density leads to universality of their local fluctuations for large M. A relation between the partial time delays and diagonal matrix elements of the Wigner-Smith matrix is revealed for ideal coupling. This helped us in deriving the joint probability distribution of partial time delays and the distribution of the Wigner time delay.
Physical Review E 04/2001; 63(3 Pt 2):035202. · 2.26 Impact Factor
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ABSTRACT: Random contractions (subunitary random matrices) appear naturally when considering quantized chaotic maps within a general
theory of open linear stationary systems with discrete time. We analyze statistical properties of complex eigenvalues of generic
N × N random matrices  of such a type, corresponding to systems with broken time reversal invariance. Deviations from unitarity are characterized
by rank M≤N and a set of eigenvalues 0<T
i≤1, i=1,..., M of the matrix
[^(T)] = [^1] - [^(A)]f [^(A)]\hat T = \hat 1 - \hat A^\dag \hat A
. We solve the problem completely by deriving the joint probability density of N complex eigenvalues and calculating all n-point correlation functions. In the limit N≫M, n, the correlation functions acquire the universal form found earlier for weakly non-Hermitian random matrices.
JETP Letters 09/2000; 72(8):422-426. · 1.35 Impact Factor
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ABSTRACT: We study complex eigenvalues of large $N\times N$ symmetric random matrices of the form ${\cal H}=\hat{H}-i\hat{\Gamma}$, where both $\hat{H}$ and $\hat{\Gamma}$ are real symmetric, $\hat{H}$ is random Gaussian and $\hat{\Gamma}$ is such that $NTr \hat{\Gamma}^2_2\sim Tr \hat{H}_1^2$ when $N\to \infty$. When $\hat{\Gamma}\ge 0$ the model can be used to describe the universal statistics of S-matrix poles (resonances) in the complex energy plane. We derive the ensuing distribution of the resonance widths which generalizes the well-known $\chi^2$ distribution to the case of overlapping resonances. We also consider a different class of "almost real" matrices when $\hat{\Gamma}$ is random and uncorrelated with $\hat{H}$. Comment: 8 pages+2 eps figures
07/1998;
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ABSTRACT: This paper is a detailed account of the recent progress in understanding the statistical properties of complex eigenvalues of random non-Hermitian matrices reported earlier in our two short communications: Physics Letters A v.226, 46 (1997) and Phys. Rev. Lett., v.79, 557 (1997)
03/1998;
