Crossover ensembles of random matrices and skew-orthogonal polynomials

Kumar, Santosh; Pandey, Akhilesh

doi:10.1016/j.aop.2011.04.013

Cited by 29 publications

(86 citation statements)

References 85 publications

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“…Just as the eigenvalue statistics for Wigner and Wishart ensembles respectively involve Hermite polynomials and associated Laguerre polynomials, the eigenvalue statistics of Jacobi ensembles involve Jacobi polynomials. Together they complete the random matrix ensemble picture in connection with the theory of classical orthogonal, and skew-orthogonal polynomials [6][7][8][9]. Closely related to these classical ensembles is the less known Cauchy-Lorentz ensemble [10,11].…”

Section: Introductionmentioning

confidence: 98%

The Correlated Jacobi and the Correlated Cauchy–Lorentz Ensembles

et al. 2015

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We calculate the k-point generating function of the correlated Jacobi ensemble using supersymmetric methods. We use the result for complex matrices for k = 1 to derive a closed-form expression for eigenvalue density. For real matrices we obtain the density in terms of a twofold integral that we evaluate numerically. For both expressions we find agreement when comparing with Monte Carlo simulations. Relations between these quantities for the Jacobi and the Cauchy-Lorentz ensemble are derived.

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Section: Introductionmentioning

confidence: 98%

The Correlated Jacobi and the Correlated Cauchy–Lorentz Ensembles

et al. 2015

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“…For instance, the ensemble averaged eigenvalue density, p(x) , is not known in a simple form [30]. Fortunately, in some of the cases we discuss below, we can approximate n-WE with that of a simpler CWE.…”

Section: Generalized Wishart Ensembles As a Model For Rdmmmentioning

confidence: 99%

Subsystem dynamics under random Hamiltonian evolution

Vinayak

Žnidarič

2012

J. Phys. A: Math. Theor.

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Abstract. We study time evolution of a subsystem's density matrix under unitary evolution, generated by a sufficiently complex, say quantum chaotic, Hamiltonian, modeled by a random matrix. We exactly calculate all coherences, purity and fluctuations. We show numerically that the reduced density matrix can be described in terms of a noncentral correlated Wishart ensemble for which we are able to perform analytical calculations of the eigenvalue density. Our description accounts for a transition from an arbitrary initial state towards a random state at large times, enabling us to determine the convergence time after which random states are reached. We identify and describe a number of other interesting features, like a series of collisions between the largest eigenvalue and the bulk, accompanied by a phase transition in its distribution function.

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“…For the above LOE to LUE crossover ensemble, the effective transition parameter is known to be λ eff ∼ �(�) [33,34,44], which depends on the location in the spectrum. However, away from the right edge, we have λ eff ∼ √ N λ and we expect this scaling to work for the the entire spectrum collectively, similar to the Gaussian case.…”

Section: Application To Laguerre Ensemblementioning

confidence: 99%

Distribution of the ratio of two consecutive level spacings in orthogonal to unitary crossover ensembles

2020

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The ratio of two consecutive level spacings has emerged as a very useful metric in exploring universal features exhibited by complex spectra. It does not require the knowledge of density of states and is therefore quite convenient to compute in studying the spectrum of a general system. The Wigner surmise like results for the ratio distribution are known for the invariant classes of Gaussian random matrices. However, for the crossover ensembles, which are useful in modeling systems with partially broken symmetries, corresponding results have remained unavailable so far. In this work, we derive exact result for the distribution of ratio of two consecutive level spacings in the Gaussian orthogonal to unitary crossover ensemble using a 3 × 3 random matrix model. This crossover is useful in modeling time reversal symmetry breaking in quantum chaotic systems. Although based on a 3 × 3 matrix model, our result can be applied in the study of large spectra also, provided the symmetry breaking parameter facilitating the crossover is suitably scaled. We substantiate this claim by considering Gaussian and Laguerre crossover ensembles comprising large matrices. Moreover, we apply our result to investigate the violation of time-reversal invariance in the quantum kicked rotor system. *

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Crossover ensembles of random matrices and skew-orthogonal polynomials

Cited by 29 publications

References 85 publications

The Correlated Jacobi and the Correlated Cauchy–Lorentz Ensembles

The Correlated Jacobi and the Correlated Cauchy–Lorentz Ensembles

Subsystem dynamics under random Hamiltonian evolution

Distribution of the ratio of two consecutive level spacings in orthogonal to unitary crossover ensembles

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