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The sum of independent Wishart matrices, taken from distributions with unequal covariance matrices, plays a crucial role in multivariate statistics, and has applications in the fields of quantitative finance and telecommunication. However, analytical results concerning the corresponding eigenvalue statistics have remained unavailable, even for the sum of two Wishart matrices. This can be attributed to the complicated and rotationally noninvariant nature of the matrix distribution that makes extracting the information about eigenvalues a nontrivial task. Using a generalization of the Harish-Chandra-Itzykson-Zuber integral, we find exact solution to this problem for the complex Wishart case when one of the covariance matrices is proportional to the identity matrix, while the other is arbitrary. We derive exact and compact expressions for the joint probability density and marginal density of eigenvalues. The analytical results are compared with numerical simulations and we find perfect agreement. © EPLA, 2014.

Physics

(SoNS) School of Natural Sciences

Fulltext

Eigenvalue statistics for the sum of two complex Wishart matrices

Scattering is a ubiquitous phenomenon which is observed in a variety of physical systems which span a wide range of length scales. The scattering matrix is the key quantity which provides a complete description of the scattering process. The universal features of scattering in chaotic systems is most generally modeled by the Heidelberg approach which introduces stochasticity to the scattering matrix at the level of the Hamiltonian describing the scattering center. The statistics of the scattering matrix is obtained by averaging over the ensemble of random Hamiltonians of appropriate symmetry. We derive exact results for the distributions of the real and imaginary parts of the off-diagonal scattering matrix elements applicable to orthogonally-invariant and unitarily-invariant Hamiltonians, thereby solving a long standing problem. © 2013 Elsevier Inc.

World University root node

Annals of Physics

Distributions of off-diagonal scattering matrix elements: Exact results

Scattering is an important phenomenon which is observed in systems ranging from the micro- to macroscale. In the context of nuclear reaction theory, the Heidelberg approach was proposed and later demonstrated to be applicable to many chaotic scattering systems. To model the universal properties, stochasticity is introduced to the scattering matrix on the level of the Hamiltonian by using random matrices. A long-standing problem was the computation of the distribution of the off-diagonal scattering-matrix elements. We report here an exact solution to this problem and present analytical results for systems with preserved and with violated time-reversal invariance. Our derivation is based on a new variant of the supersymmetry method. We also validate our results with scattering data obtained from experiments with microwave billiards. © 2013 American Physical Society.

Physical Review Letters

Distribution of scattering matrix elements in quantum chaotic scattering

We propose a method based on the Wang-Landau algorithm to numerically generate the spectral densities of random matrix ensembles. The method employs Dyson's log-gas formalism for random matrix eigenvalues and also enables one to simultaneously investigate the thermodynamic properties. This approach is a powerful alternative to the conventionally used Monte Carlo simulations based on the Boltzmann sampling, and is ideally suited for investigating β-ensembles. Copyright © EPLA, 2013.

Random matrix ensembles: Wang-Landau algorithm for spectral densities

In a recent paper (S. Kumar, A. Pandey, Phys. Rev. E, 79, 2009, p. 026211) we considered Jacobi family (including Laguerre and Gaussian cases) of random matrix ensembles and reported exact solutions of crossover problems involving time-reversal symmetry breaking. In the present paper we give details of the work. We start with Dyson's Brownian motion description of random matrix ensembles and obtain universal hierarchic relations among the unfolded correlation functions. For arbitrary dimensions we derive the joint probability density (jpd) of eigenvalues for all transitions leading to unitary ensembles as equilibrium ensembles. We focus on the orthogonal-unitary and symplectic-unitary crossovers and give generic expressions for jpd of eigenvalues, two-point kernels and n-level correlation functions. This involves generalization of the theory of skew-orthogonal polynomials to crossover ensembles. We also consider crossovers in the circular ensembles to show the generality of our method. In the large dimensionality limit, correlations in spectra with arbitrary initial density are shown to be universal when expressed in terms of a rescaled symmetry breaking parameter. Applications of our crossover results to communication theory and quantum conductance problems are also briefly discussed. © 2011 Elsevier Inc.

Crossover ensembles of random matrices and skew-orthogonal polynomials

Quantum entanglement plays a crucial role in quantum information, quantum teleportation and quantum computation. The information about the entanglement content between subsystems of the composite system is encoded in the Schmidt eigenvalues. We derive here closed expressions for the spectral density of Schmidt eigenvalues for all three invariant classes of random matrix ensembles. We also obtain exact results for average von Neumann entropy. We find that maximum average entanglement is achieved if the system belongs to the symplectic invariant class. © 2011 IOP Publishing Ltd.

Journal of Physics A: Mathematical and Theoretical

Entanglement in random pure states: Spectral density and average von Neumann entropy

We consider the conductance distributions in chaotic mesoscopic cavities for all three invariant classes of random matrices for the arbitrary number of channels N1, N2 in the connecting leads. We show that the Laplace transforms of the distributions can be expressed in terms of determinants in the unitary case and Pfaffians in the orthogonal and symplectic cases. The inverse Laplace transforms then give the exact distributions. This formalism is particularly useful for small values of N = min (N1, N2), and thus is of direct experimental relevance. We also obtain the conductance distributions for orthogonal-unitary and symplectic-unitary crossover ensembles. © 2010 IOP Publishing Ltd.

Conductance distributions in chaotic mesoscopic cavities

We study the transition in conductance properties of chaotic mesoscopic cavities as time-reversal symmetry is broken. We consider the Brownian motion model for transmission eigenvalues for both types of transitions, namely, orthogonal-unitary and symplectic-unitary crossovers depending on the presence or absence of spin-rotation symmetry of the electron. In both cases the crossover is governed by a Brownian motion parameter τ, which measures the extent of time-reversal symmetry breaking. It is shown that the results obtained correspond to the Jacobi crossover ensembles of random matrices. We derive the level density and the correlation functions of higher orders for the transmission eigenvalues. We also obtain the exact expressions for the average conductance, average shot-noise power and variance of conductance, as functions of τ, for the arbitrary number of modes (channels) in the two leads connected to the cavity. Moreover, we give the asymptotic result for the variance of shot-noise power for both the crossovers, the exact results being too long. In the τ → 0 and τ → ∞ limits, the known results for the orthogonal (or symplectic) and unitary ensembles are reproduced. In the weak time-reversal symmetry breaking regime, our results are shown to be in agreement with the semiclassical predictions. © 2010 IOP Publishing Ltd.

Jacobi crossover ensembles of random matrices and statistics of transmission eigenvalues

Random matrix model for the Nakagami-q (Hoyt) fading in multiple-input multiple-output (MIMO) communication channels with arbitrary number of transmitting and receiving antennas is considered. The joint probability density for the eigenvalues of H† H† (or {H H†), where H† is the channel matrix, is shown to correspond to the Laguerre crossover ensemble of random matrices and is given in terms of a Pfaffian. Exact expression for the marginal density of eigenvalues is obtained as a series consisting of associated Laguerre polynomials. This is used to study the effect of fading on the Shannon channel capacity. Exact expressions for higher order density correlation functions are also given which can be used to study the distribution of channel capacity. © 2010 IEEE.}, author_keywords={Channel capacity; Fading distributions; Hoyt distribution; Laguerre crossover ensemble; Multiple-input multiple-output (MIMO) channels; Nakagami-q distribution; Random matrices}, keywords={Hoyt distribution; Laguerre; Multiple-input multiple-output channels; Nakagami; Nakagami-q distribution; Random matrices, Distribution functions; Eigenvalues and eigenfunctions; Fading channels; Probability density function; Receiving antennas, Channel capacity}, funding_details={Council of Scientific and Industrial Research, IndiaCouncil of Scientific and Industrial Research, India, CSIR}, funding_text_1={Manuscript received August 03, 2009; revised December 08, 2009. Current version published April 21, 2010. S. Kumar was supported by CSIR India through a fellowship. The authors are with the School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India (e-mail: skumar.physics@gmail.com; ap0700@mail.jnu.ac.in). Communicated by A. L. Moustakas, Associate Editor for Communications. Digital Object Identifier 10.1109/TIT.2010.2044060}, references={Simon, M.K., Alouini, M.-S., (2000) Digital Communication Over Fading Channels: A Unified Approach to Performance Analysis, , Hoboken, NJ: Wiley; Simon, M.K., Alouini, M.-S., A unified approach to the performance analysis of digital communication over generalized fading channels (1998) Proc. IEEE, 86 (9), pp. 1860-1877. , Sep; Foschini, G.J., Gans, M.J., On limits of wireless communications in a fading environment when using multiple antennas (1998) Wireless Pers. Commun., 6 (2), pp. 311-335. , Mar; Telatar, I.E., Capacity of multi-antenna Gaussian channels (1999) Eur. Trans. 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Theory, 52 (12), pp. 5336-5351. , Dec; Ghaderipoor, A., Tellambura, C., Noori, M., On the eigenvalue distribution of correlated MIMO channels by character expansion of groups (2008) Proc. IEEE Global Telecommunications Conf, pp. 1-5. , Dec; Marques, P.M., Abrantes, S.A., On the derivation of the exact, closed-form capacity formulas for receiver-sided correlated MIMO channels (2008) IEEE Trans. Inf. Theory, 54 (3), pp. 1139-1161. , Mar; Kang, M., Alouini, M.-S., Quadratic forms in complex Gaussian matrices and performance analysis of MIMO systems with co-channel interference (2004) IEEE Trans. Wireless Commun., 3 (2), pp. 418-431. , Mar; Ordôñez, L.G., Palomar, D.P., Fonollosa, J.R., Ordered eigenvalues of a general class of Hermitian random matrices and performance analysis of MIMO systems (2008) Proc. IEEE Conf. 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Communications, 1, pp. 406-410; Ge, H., Wong, K.D., Barton, M., Liberti, J.C., Statistical characterization of MIMO channel capacity (2002) Proc. IEEE Wireless Communications and Networking Conf, 2, pp. 789-793. , Mar; Kamath, M.A., Hughes, B.L., Yu, X., Gaussian approximations for the capacity of MIMO Rayleigh fading channels (2002) Proc. 36th Asilomar Conf. Signals, Systems and Computers, 1, pp. 614-618. , Nov; Zhao, Y., Zhao, M., Xiao, L., Wang, J., Analytical expression for the MIMO channel capacity (2006) Tsinghua Sci. Technol., 11 (3), pp. 271-277. , Jun; Chizhik, D., Foschini, G., Gans, M., Valenzela, R., Keyholes, correlations, and capacities of multielement transmit and receive antennas (2002) IEEE Trans. Wireless Commun., 1 (2), pp. 361-368. , Apr; Shin, H., Lee, J.H., Capacity of multiple-antenna fading channels: Spatial fading correlation, double scattering, and keyholes (2003) IEEE Trans. Inf. Theory, 49 (10), pp. 2636-2647. , Oct; Wishart, J.B., (1928), 20 A, pp. 32-52; Wilks, S.M., (1963) Mathematical Statistics, , Hoboken, NJ: Wiley; Edelman, A., (1989) Eigenvalues and Condition Numbers of Random Matrices, , Ph. D. dissertation, Dept. Math., Massachusetts Inst. Technol., Cambridge, MA; Mehta, M.L., (2004) Random Matrices, , New York: Academic; Pandey, A., Ghosh, S., Skew-orthogonal polynomials and universality of energy-level correlations (2001) Phys. Rev. Lett., 87 (2), p. 02410214. , Jun; Ghosh, S., Pandey, A., Skew-orthogonal polynomials and randommatrix ensembles (2002) Phys. Rev. E, 65 (1-21), p. 046221. , Apr; Dyson, F.J., Distribution of eigenvalues for a class of real symmetric matrices (1971) Rev. Mex. Fis., 20, pp. 231-237; Brody, T.A., Flores, J., French, J.B., Mello, P.A., Pandey, A., Wong, S.S.M., Random-matrix physics: Spectrum and strength fluctuations (1981) Rev. Mod. 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IEEE Transactions on Information Theory

Random matrix model for nakagamihoyt fading

Orthogonal-unitary and symplectic-unitary crossover ensembles of random matrices are relevant in many contexts, especially in the study of time reversal symmetry breaking in quantum chaotic systems. Using skew-orthogonal polynomials we show that the same generic form of n -level correlation functions are obtained for the Jacobi family of crossover ensembles, including the Laguerre and Gaussian cases. For large matrices we find universal forms of unfolded correlation functions when expressed in terms of a rescaled transition parameter with arbitrary initial level density. © 2009 The American Physical Society.