Selberg Integrals and Hypergeometric Functions Associated with Jack Polynomials

Kaneko, Jyoichi

doi:10.1137/0524064

Cited by 205 publications

(272 citation statements)

References 14 publications

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“…If α is zero or an odd integer, RMT results for the distribution of the smallest eigenvalue are known analytically [15,16]. For even α = 0 (this is relevant in our case), they can be obtained as an expansion in zonal polynomials [17,12]. For µ = 0, the RMT results for ρ s (z) and P (λ min ) can be obtained numerically by constructing skew-orthogonal polynomials which obey orthogonality relations determined by a weight function involving the fermion determinants [14,12].…”

Section: Results With Dynamical Fermionsmentioning

confidence: 99%

Universal and non-universal behavior in Dirac spectra

Berbenni-Bitsch

Göckeler

Meyer

et al. 1999

Nuclear Physics B - Proceedings Supplements

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We have computed ensembles of complete spectra of the staggered Dirac operator using four-dimensional SU(2) gauge fields, both in the quenched approximation and with dynamical fermions. To identify universal features in the Dirac spectrum, we compare the lattice data with predictions from chiral random matrix theory for the distribution of the low-lying eigenvalues. Good agreement is found up to some limiting energy, the so-called Thouless energy, above which random matrix theory no longer applies. We determine the dependence of the Thouless energy on the simulation parameters using the scalar susceptibility and the number variance.

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Section: Results With Dynamical Fermionsmentioning

confidence: 99%

Universal and non-universal behavior in Dirac spectra

Berbenni-Bitsch

Göckeler

Meyer

et al. 1999

Nuclear Physics B - Proceedings Supplements

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“…Fortunately, the averages of Jack polynomials in the Selberg model are well-known to be simple quantities [117,118]. They factorize nicely, and can be generally expressed by the Kadell formula…”

Section: Eq (120) Via Jack Expansionmentioning

confidence: 99%

Brezin-Gross-Witten model as “pure gauge” limit of Selberg integrals

Миронов

Морозов

Shakirov

2011

J. High Energ. Phys.

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Abstract:The AGT relation identifies the Nekrasov functions for various N = 2 SUSY gauge theories with the 2d conformal blocks, which possess explicit Dotsenko-Fateev matrix model (β-ensemble) representations the latter being polylinear combinations of Selberg integrals. The "pure gauge" limit of these matrix models is, however, a non-trivial multiscaling large-N limit, which requires a separate investigation. We show that in this pure gauge limit the Selberg integrals turn into averages in a Brezin-Gross-Witten (BGW) model. Thus, the Nekrasov function for pure SU(2) theory acquires a form very much reminiscent of the AMM decomposition formula for some model X into a pair of the BGW models. At the same time, X, which still has to be found, is the pure gauge limit of the elliptic Selberg integral. Presumably, it is again a BGW model, only in the Dijkgraaf-Vafa double cut phase.

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“…Identity (16) and its dual version (21) are rather useful in the context of Schur function expansions. For example, the matrix integrals (25) and (26) are straightforward corollaries of these identities. Consider, for example, the matrix integral in (25).…”

Section: Identity (17) Impliesmentioning

confidence: 99%

“…For example, the matrix integrals (25) and (26) are straightforward corollaries of these identities. Consider, for example, the matrix integral in (25). Expanding each of the determinants on the left-hand side with the help of (6), one arrive by the way of the orthogonality relation (58) at…”

Section: Identity (17) Impliesmentioning

confidence: 99%

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A few remarks on colour–flavour transformations, truncations of random unitary matrices, Berezin reproducing kernels and Selberg-type integrals

Fyodorov¹,

Khoruzhenko²

2007

J. Phys. A: Math. Theor.

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We investigate diverse relations of the colour-flavour transformations (CFT) introduced by Zirnbauer in [41,42] to various topics in random matrix theory and multivariate analysis, such as measures on truncations of unitary random matrices, Jacobi ensembles of random matrices, Berezin reproducing kernels and a generalization of the Selberg integral due to Kaneko, Kadell and Yan involving the Schur functions. Apart from suggesting explicit formulas for bosonic CFT for the unitary group in the range of parameters beyond that in [42], we also suggest an alternative variant of the transformation, with integration going over an unbounded domain of a pair of Hermitian matrices. The latter makes possible evaluation of certain averages in random matrix theory.

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Selberg Integrals and Hypergeometric Functions Associated with Jack Polynomials

Cited by 205 publications

References 14 publications

Universal and non-universal behavior in Dirac spectra

Universal and non-universal behavior in Dirac spectra

Brezin-Gross-Witten model as “pure gauge” limit of Selberg integrals

A few remarks on colour–flavour transformations, truncations of random unitary matrices, Berezin reproducing kernels and Selberg-type integrals

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