A Fuchsian Matrix Differential Equation for Selberg Correlation Integrals

Forrester, Peter J.; Rains, Eric M.

doi:10.1007/s00220-011-1305-y

Cited by 20 publications

(24 citation statements)

References 38 publications

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“…Recursive relations from the broader theory of the Selberg integral (see chapter 4 in Ref. ) allow for the computation of averages of the form

{(〈〉, \prod_{l = 1}^{N}, {(x - x_{l})}^{n})}_{L β E (x^{a} e^{- β x / 2})},

where

{false⟨ \cdot false⟩}_{L β E false(x^{a} e^{- β x / 2} false)}

denotes the expectation with respect to the PDF .…”

Section: Numerics Beyond β=2 General a And A=1 General βmentioning

confidence: 99%

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Finite‐size corrections at the hard edge for the Laguerre β ensemble

Forrester

Trinh

2019

Stud Appl Math

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A fundamental question in random matrix theory is to quantify the optimal rate of convergence to universal laws. We take up this problem for the Laguerre β ensemble, characterized by the Dyson parameter β, and the Laguerre weight xae−βx/2, x>0 in the hard edge limit. The latter relates to the eigenvalues in the vicinity of the origin in the scaled variable x↦x/4N. Previous work has established the corresponding functional form of various statistical quantities—for example, the distribution of the smallest eigenvalue, provided that a∈double-struckZ≥0. We show, using the theory of multidimensional hypergeometric functions based on Jack polynomials, that with the modified hard edge scaling x↦x/4(N+a/β), the rate of convergence to the limiting distribution is O(1/N2), which is optimal. In the case β=2, general a>−1 the explicit functional form of the distribution of the smallest eigenvalue at this order can be computed, as it can for a=1 and general β>0. An iterative scheme is presented to numerically approximate the functional form for general a∈double-struckZ≥2.

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“…Recursive relations from the broader theory of the Selberg integral (see chapter 4 in Ref. ) allow for the computation of averages of the form

{(〈〉, \prod_{l = 1}^{N}, {(x - x_{l})}^{n})}_{L β E (x^{a} e^{- β x / 2})},

where

{false⟨ \cdot false⟩}_{L β E false(x^{a} e^{- β x / 2} false)}

denotes the expectation with respect to the PDF .…”

Section: Numerics Beyond β=2 General a And A=1 General βmentioning

confidence: 99%

“…Recursive relations [38][39][40] from the broader theory of the Selberg integral (see chapter 4 in Ref. 22) allow for the computation of averages of the form…”

Section: Numerics Beyond = General and = Generalmentioning

confidence: 99%

Finite‐size corrections at the hard edge for the Laguerre β ensemble

Forrester

Trinh

2019

Stud Appl Math

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“…Our motivation arises from the fact that the structure of zeros of certain Wronskians of orthogonal polynomials or special functions has recently been studied numerically (cf. [19,25,27] and the references therein), where it is shown that they have highly regular configurations in the complex plane. It turns out that the zeros of Wronskians for certain multiple orthogonal polynomials produce intriguing pictures as well, which might be of independent interest.…”

Section: Conf Igurations Of Zeros For the Wronskians Of Multiple Hermmentioning

confidence: 97%

On Certain Wronskians of Multiple Orthogonal Polynomials

Zhang¹

2014

SIGMA

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Abstract. We consider determinants of Wronskian type whose entries are multiple orthogonal polynomials associated with a path connecting two multi-indices. By assuming that the weight functions form an algebraic Chebyshev (AT) system, we show that the polynomials represented by the Wronskians keep a constant sign in some cases, while in some other cases oscillatory behavior appears, which generalizes classical results for orthogonal polynomials due to Karlin and Szegő. There are two applications of our results. The first application arises from the observation that the m-th moment of the average characteristic polynomials for multiple orthogonal polynomial ensembles can be expressed as a Wronskian of the type II multiple orthogonal polynomials. Hence, it is straightforward to obtain the distinct behavior of the moments for odd and even m in a special multiple orthogonal ensemble -the AT ensemble. As the second application, we derive some Turán type inequalities for multiple Hermite and multiple Laguerre polynomials (of two kinds). Finally, we study numerically the geometric configuration of zeros for the Wronskians of these multiple orthogonal polynomials. We observe that the zeros have regular configurations in the complex plane, which might be of independent interest.

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“…. This family of multiple integrals satisfies the differentialdifference system [17], [19, §4.6.4] 1 , later observed to be equivalent to a certain Fuchsian matrix differential equation [26],…”

Section: Introductionmentioning

confidence: 99%

Computable structural formulas for the distribution of the $β$-Jacobi edge eigenvalues

Forrester¹,

Kumar²

2020

Preprint

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The Jacobi ensemble is one of the classical ensembles of random matrix theory. Prominent in applications are properties of the eigenvalues at the spectrum edge, specifically the distribution of the largest (e.g. Roy's largest root test in multivariate statistics) and smallest (e.g. condition numbers of linear systems) eigenvalues. We identify three ranges of parameter values for which the gap probability determining these distributions is a finite sum with respect to particular bases, and moreover make use of a certain differential-difference system fundamental in the theory of the Selberg integral to provide a recursive scheme to compute the corresponding coefficients.

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A Fuchsian Matrix Differential Equation for Selberg Correlation Integrals

Cited by 20 publications

References 38 publications

Finite‐size corrections at the hard edge for the Laguerre β ensemble

Finite‐size corrections at the hard edge for the Laguerre β ensemble

On Certain Wronskians of Multiple Orthogonal Polynomials

Computable structural formulas for the distribution of the $β$-Jacobi edge eigenvalues

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