Skew-orthogonal polynomials and random-matrix ensembles

Ghosh, Saugata; Pandey, Akhilesh

doi:10.1103/physreve.65.046221

Cited by 33 publications

(168 citation statements)

References 18 publications

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“…Hence we concentrate on the calculation of the level density (7). Already computing the level density with the help of the projection formula is quite involved.…”

Section: Eigenvalue Density In the Real Ensemblementioning

confidence: 99%

“…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: 99%

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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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“…Hence we concentrate on the calculation of the level density (7). Already computing the level density with the help of the projection formula is quite involved.…”

Section: Eigenvalue Density In the Real Ensemblementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Correlated Jacobi and the Correlated Cauchy–Lorentz Ensembles

et al. 2015

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“…(25), (26) and the asymptotic results for these SOP [5], we present an alternative derivation of the level-density and the 'two-point' function for the Gaussian orthogonal and symplectic ensembles. For the Gaussian Ensembles, with d = 1, (25) and (26) give…”

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confidence: 99%

“…Some progress has been made in this regard [1,4,5] to develop the theory of SOP. Alternative approaches were taken by various authors [6,7,8,9,10,11,12,13] to study these ensembles.…”

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Generalized Christoffel–Darboux formula for skew-orthogonal polynomials and random matrix theory

Ghosh¹

2006

J. Phys. A: Math. Gen.

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We obtain a generalized Christoffel-Darboux (GCD) formula for skew-orthogonal polynomials. Using this, we present an alternative derivation of the level density and two-point function for Gaussian orthogonal ensembles and Gaussian symplectic ensembles of random matrices.PACS numbers: 02.30. Gp, 05.45.Mt Random matrices have found applications in different branches of physics mainly due to the 'universality' in their correlation function under certain scaling limits. In this context, although unitary ensembles of random matrices, corresponding to systems with broken time-reversal symmetry (for example, a mesoscopic conductor in the presence of a magnetic field) have been extensively studied [1,2], much less is known about Orthogonal Ensembles (OE) and Symplectic Ensembles (SE) of random matrices.In his phenomenal paper [3] Dyson stressed the importance of a good understanding of the skew-orthogonal polynomials (SOP) to study the OE and SE of random matrices. Some progress has been made in this regard [1,4,5] to develop the theory of SOP. Alternative approaches were taken by various authors [6,7,8,9,10,11,12,13] to study these ensembles. For example, Deift and Gioev [14,15] have recently used the Widom's representation [16] to prove Universality for a wide class of polynomial potentials of the OE and SE of random matrices. In all these work, the authors have used the well known properties of orthogonal polynomials to study these ensembles. Here we use the more elegant SOP, evolving naturally from OE and SE of random matrices.In this paper we obtain recursion relations for SOP. Using this we derive the GCD formula. We use them to obtain the level density and two-point correlation function for Gaussian orthogonal ensembles and Gaussian symplectic ensembles of random matrices.We consider ensembles of 2N dimensional matrices H with probability distributionwhere the parameter β = 1 and 4 corresponds to H real symmetric or quaternion real self dual.(Note that we have considered the OE of even dimension. The odd dimension can be easily generalized.) dH is the standard Haar measure. Z βN is the so called 'partition function', and is proportional to the product of skew-normalization constants [1]:Here n F is called the 'Fermi level' by analogy with a system of fermions. In our case, it takes the value n F = 2N for β = 1 and 4 respectively. It is assumed that most of the physical quantities (like level density and correlation functions) are related to properties of g (β)

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