Skew-Orthogonal Polynomials and Random Matrix
                    Theory

Ghosh, Saugata

doi:10.1090/crmm/028

Cited by 7 publications

(21 citation statements)

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“…These three relations are already well known [4,48]. They can also be derived with help of the Christoffel-Darboux formula.…”

Section: Review Of Chiral Unitary Random Matricesmentioning

confidence: 84%

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Surprising Pfaffian factorizations in random matrix theory with Dyson index β = 2

Kieburg

2012

J. Phys. A: Math. Theor.

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Abstract. In the past decades, determinants and Pfaffians were found for eigenvalue correlations of various random matrix ensembles. These structures simplify the average over a large number of ratios of characteristic polynomials to integrations over one and two characteristic polynomials only. Up to now it was thought that determinants occur for ensembles with Dyson index β = 2 whereas Pfaffians only for ensembles with β = 1, 4. We derive a non-trivial Pfaffian determinant for β = 2 random matrix ensembles which is similar to the one for β = 1, 4. Thus, it unveils a hidden universality of this structure. We also give a general relation between the orthogonal polynomials related to the determinantal structure and the skew-orthogonal polynomials corresponding to the Pfaffian. As a particular example we consider the chiral unitary ensembles in great detail.

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“…These three relations are already well known [4,48]. They can also be derived with help of the Christoffel-Darboux formula.…”

Section: Review Of Chiral Unitary Random Matricesmentioning

confidence: 84%

“…[34,35] circumvents the integration theorem by Dyson and Metha [24,25,4,48]. Moreover the approach of Refs.…”

Section: Preliminariesmentioning

confidence: 99%

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Surprising Pfaffian factorizations in random matrix theory with Dyson index β = 2

Kieburg

2012

J. Phys. A: Math. Theor.

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“…Exactly at the critical value, two of these bands coincide to form an upper band, and the third band forms a lower band, creating a pseudo two band structure. X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X In this subsection, we derive the level density using the method of resolvent [3,10] as N → ∞. The result is expressed in terms of moments M k (1.5) which are derived using results from the Freud's equation.…”

Section: Critical A4'smentioning

confidence: 99%

“…We then compare the result with the level density for N = 30 using (1.4). Here, we recall [3]that we use the scaling V (x) → V (x)/2 to obtain the corresponding results.…”

Section: Critical A4'smentioning

confidence: 99%

Generalized Freud’s equation and level densities with polynomial potential

Boobna¹,

Ghosh²

2013

Pramana - J Phys

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We study orthogonal polynomials with weight exp[−N V (x)], where2k /2k is a polynomial of order 2d. We derive the generalised Freud's equations for d = 3, 4 and 5 and using this obtain Rµ = hµ/hµ−1, where hµ is the normalization constant for the corresponding orthogonal polynomials. Moments of the density functions, expressed in terms of Rµ, are obtained using Freud's equation and using this, explicit results of level densities as N → ∞ are derived.

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Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices

Akemann

Kieburg

Phillips

2010

J. Phys. A: Math. Theor.

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We apply the method of skew-orthogonal polynomials (SOP) in the complex plane to asymmetric random matrices with real elements, belonging to two different classes. Explicit integral representations valid for arbitrary weight functions are derived for the SOP and for their Cauchy transforms, given as expectation values of traces and determinants or their inverses, respectively. Our proof uses the fact that the joint probability distribution function for all combinations of real eigenvalues and complex conjugate eigenvalue pairs can be written as a product. Examples for the SOP are given in terms of Laguerre polynomials for the chiral ensemble (also called the non-Hermitian real Wishart-Laguerre ensemble), both without and with the insertion of characteristic polynomials. Such characteristic polynomials play the role of mass terms in applications to complex Dirac spectra in field theory. In addition, for the elliptic real Ginibre ensemble we recover the SOP of Forrester and Nagao in terms of Hermite polynomials.

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Skew-Orthogonal Polynomials and Random Matrix Theory

Cited by 7 publications

References 0 publications

Surprising Pfaffian factorizations in random matrix theory with Dyson index β = 2

Surprising Pfaffian factorizations in random matrix theory with Dyson index β = 2

Generalized Freud’s equation and level densities with polynomial potential

Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices

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