Correlation Functions of Random Matrix EnsemblesRelated to Classical Orthogonal Polynomials

Nagao, Taro; Wadati, Miki

doi:10.1143/jpsj.60.3298

Cited by 97 publications

(93 citation statements)

References 19 publications

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“…Moreover, it is well-known that the limiting behavior of K n is given by the sine kernel S(x, y) := sin π(x − y) x − y (1. 8) in the bulk of the spectrum [20,33], by the Airy kernel…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

Dai

Zhao

2014

Commun. Math. Phys.

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In this paper, we study the singularly perturbed Laguerre unitary ensemblewith V t (x) = x + t/x, x ∈ (0, +∞) and t > 0. Due to the effect of t/x for varying t, the eigenvalue correlation kernel has a new limit instead of the usual Bessel kernel at the hard edge 0. This limiting kernel involves ψ-functions associated with a special solution to a new thirdorder nonlinear differential equation, which is then shown equivalent to a particular Painlevé III equation. The transition of this limiting kernel to the Bessel and Airy kernels is also studied when the parameter t changes in a finite interval (0, d]. Our approach is based on Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems.

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“…Moreover, it is well-known that the limiting behavior of K n is given by the sine kernel S(x, y) := sin π(x − y) x − y (1. 8) in the bulk of the spectrum [20,33], by the Airy kernel…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…[10,13]), it follows from (3.32) that 33) uniformly for z in the whole complex plane. This completes the nonlinear steepest descent analysis.…”

Section: The Final Transformation S → Rmentioning

confidence: 99%

Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

Dai

Zhao

2014

Commun. Math. Phys.

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“…From the formulas of Laguerre and Hermite polynomial, Forrester [25] obtains the Bessel kernel and the Airy kernel after suitable re-scaling. Moreover, Nagao and Wadati [44] deduced the Bessel kernel by scaling the Jacobi ensemble at the hard edges, ±1 . Kuijlaars and Zhang [38] obtain a limiting kernel as a generalization of Bessel kernel by scaling the correlation kernel of complex Ginibre random matrices at the hard edge, see the references therein for more information.…”

Section: An Equivalent Expression Readsmentioning

confidence: 99%

“…The series expansion of the Hankel determinant is given by, 44) subject to α / ∈ Z, α > 0 and | arg t| < π. D n (0, α, β) has a closed form expression [4],…”

Section: )mentioning

confidence: 99%

Perturbed Hankel determinant, correlation functions and Painlevé equations

Chen¹,

Chen²,

Fan

2016

Journal of Mathematical Physics

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We continue with the study of the Hankel determinant, D n (t, α, β) := det , generated by a Pollaczek-Jacobi type weight,This reduces to the "pure" Jacobi weight at t = 0. We may take α ∈ R , in the situation while t is strictly greater than 0. It was shown in Chen and Dai (2010), that the logarithmic derivative of this Hankel determinant satisfies a Jimbo-Miwa-Okamoto σ -form of Painlevé V ( P V ). In fact the logarithmic of the Hankel determinant has an integral representation in terms of a particular P V . In this paper, we show that, under a double scaling, where n the dimension of the Hankel matrix tends to ∞ , and t tends to 0 + , such that s := 2n 2 t is finite, the double scaled Hankel determinant (effectively an operator determinant) has an integral representation in terms of a particular P III ′ . Expansions of the scaled Hankel determinant for small and large s are found. A further double scaling with α = −2n + λ, where n → ∞ and t, tends to 0 + , such that s := nt is finite. In this situation the scaled Hankel determinant has an integral representation in terms of a particular P V , and its small and large s asymptotic expansions are also found. The reproducing kernel in terms of monic polynomials orthogonal with respect to the Pollaczek-Jacobi type weight, under the origin (or hard edge) scaling may be expressed in terms of the solutions of a second order linear ordinary differential equation (ODE). * chenminfst@gmail.com † Corresponding author(Yang Chen), yayangchen@umac.mo and yangbrookchen@yahoo.co.uk ‡ faneg@fudan.edu.cn 1 With special choices of the parameters, the limiting (double scaled) kernel and the second order ODE degenerate to Bessel kernel and the Bessel differential equation, respectively. We also applied this method to polynomials orthogonal with respect to the perturbed Laguerre weight; w(x; t, α) := x α e −x e −t/x , 0 ≤ x < ∞, α > 0, t > 0. The scaled kernel at origin of this perturbed Laguerre ensemble has the same behavior with the above limiting kernel, although difference scaled schemes are adopted on these two kernels.2

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“…The chiral ensembles with real or quaternion real matrix elements (known as the chiral Orthogonal Ensemble (chOE) and the chiral Symplectic Ensemble (chSE), respectively) are mathematically much more complicated. The general result for the microscopic spectral density of the chiral Gaussian Orthogonal Ensemble (chGOE) [34] and the of the chiral Gaussian Symplectic Ensemble (chGSE) [35] was first obtained by an explicit construction of the corresponding skew-orthogonal polynomials [36,6,37] (several special cases were analyzed in [38,20,39]). This method does not seem to be easy generalizable to non-Gaussian probability potentials.…”

Section: Introductionmentioning

confidence: 99%

Spectral universality of real chiral random matrix ensembles

Klein

Verbaarschot

2000

Nuclear Physics B

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We investigate the universality of microscopic eigenvalue correlations for Random Matrix Theories with the global symmetries of the QCD partition function. In this article we analyze the case of real valued chiral Random Matrix Theories (β = 1) by relating the kernel of the correlations functions for β = 1 to the kernel of chiral Random Matrix Theories with complex matrix elements (β = 2), which is already known to be universal. Our proof is based on a novel asymptotic property of the skew-orthogonal polynomials: an integral over the corresponding wavefunctions oscillates about half its asymptotic value in the region of the bulk of the zeros. This results solves the puzzle that microscopic universality persists in spite of contributions to the microscopic correlators from the region near the largest zero of the skew-orthogonal polynomials. Our analytical results are illustrated by the numerical construction of the skew-orthogonal polynomials for an x 4 probability potential.

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Correlation Functions of Random Matrix EnsemblesRelated to Classical Orthogonal Polynomials

Cited by 97 publications

References 19 publications

Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

Perturbed Hankel determinant, correlation functions and Painlevé equations

Spectral universality of real chiral random matrix ensembles

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