Jacobi Polynomials Associated with Selberg Integrals

Aomoto, Kazuhiko

doi:10.1137/0518042

Cited by 143 publications

(148 citation statements)

References 5 publications

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“…The natural question is whether such models exist for the last member of the classical triplet, Jacobi. The β-Jacobi ensembles have been intensively studied as theoretical distributions, especially in connection with Selberg-type integrals and Jack (or Jack-Selberg) polynomials (see [1], [15], [16], [3]). Finding a random matrix model that corresponds to them would be of much interest.…”

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confidence: 99%

Matrix models for beta ensembles

Dumitriu¹,

Edelman²

2002

Journal of Mathematical Physics

557

822

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This paper constructs tridiagonal random matrix models for general (β > 0) β-Hermite (Gaussian) and β-Laguerre (Wishart) ensembles. These generalize the well-known Gaussian and Wishart models for β = 1, 2, 4. Furthermore, in the cases of the β-Laguerre ensembles, we eliminate the exponent quantization present in the previously known models.We further discuss applications for the new matrix models, and present some open problems.

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mentioning

confidence: 99%

Matrix models for beta ensembles

Dumitriu¹,

Edelman²

2002

Journal of Mathematical Physics

557

822

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“…Up to now, we have been unable unable to find a hyperdeterminantal interpretation of this. In the same way, Anderson's (see [1]) and Aomoto's (see [3]) proofs of Selberg's integral seem to contain information unrelated to the hyperdeterminantal representations.…”

Section: Resultsmentioning

confidence: 93%

Hyperdeterminantal calculations of Selberg's and Aomoto's integrals

Luque¹,

Thibon²

2004

Molecular Physics

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The hyperdeteminants considered here are the simplest analogues of determinants for higher rank tensors which have been defined by Cayley, and apply only to tensors with an even number of indices. We have shown in a previous article that the calculation of certain multidimensional integrals could be reduced to the evaluation of hyperdeterminants of Hankel type. Here, we carry out this computation by purely algebraic means in the cases of Selberg's and Aomoto's integrals.

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“…Let ρ (1) We are motivated by some earlier work of one of the present authors and collaborators [23]. That work relates to the bulk scaled limit of the two-point correlation function for the circular ensemble (1.4) with V (θ) independent of θ.…”

Section: Jhep02(2015)173mentioning

confidence: 99%

“…We will assume these conditions henceforth. Our approach is an adaptation of Aomoto's method [1], which is also detailed in depth in Chapter 4.6 of [22].…”

Section: Jhep02(2015)173mentioning

confidence: 99%

Loop equation analysis of the circular β ensembles

Witte

Forrester

2015

J. High Energ. Phys.

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We construct a hierarchy of loop equations for invariant circular ensembles. These are valid for general classes of potentials and for arbitrary inverse temperatures Re β > 0 and number of eigenvalues N . Using matching arguments for the resolvent functions of linear statistics f (ζ) = (ζ + z)/(ζ − z) in a particular asymptotic regime, the global regime, we systematically develop the corresponding large N expansion and apply this solution scheme to the Dyson circular ensemble. Currently we can compute the second resolvent function to ten orders in this expansion and also its general Fourier coefficient or moment m k to an equivalent length. The leading large N , large k, k/N fixed form of the moments can be related to the small wave-number expansion of the structure function in the bulk, scaled Dyson circular ensemble, known from earlier work. From the moment expansion we conjecture some exact partial fraction forms for the low k moments. For all of the forgoing results we have made a comparison with the exactly soluble cases of β = 1, 2, 4, general N and even, positive β, N = 2, 3.

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Jacobi Polynomials Associated with Selberg Integrals

Cited by 143 publications

References 5 publications

Matrix models for beta ensembles

Matrix models for beta ensembles

Hyperdeterminantal calculations of Selberg's and Aomoto's integrals

Loop equation analysis of the circular β ensembles

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