Continuous and Discrete Painlevé Equations Arising from the Gap Probability Distribution of the Finite $$n$$ n Gaussian Unitary Ensembles

Cao, Mengji; Chen, Yang; Griffin, James

doi:10.1007/s10955-014-1076-x

Cited by 8 publications

(12 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [5], such an approach was taken to study the gap probability problem for the Gaussian unitary ensembles (the symmetric situation), namely, the probability that the interval J := (−a, a) is free of eigenvalues. Unfortunately the authors have made a mistake: One term was missed in an equation obtained from the sum-rule.…”

mentioning

confidence: 99%

Asymptotic gap probability distributions of the Gaussian unitary ensembles and Jacobi unitary ensembles

2018

Self Cite

View full text Add to dashboard Cite

In this paper, we address a class of problems in unitary ensembles. Specifically, we study the probability that a gap symmetric about 0, i.e. (−a, a) is found in the Gaussian unitary ensembles (GUE) and the Jacobi unitary ensembles (JUE) (where in the JUE, we take the parameters α = β). By exploiting the even parity of the weight, a doubling of the interval to (a 2 , ∞) for the GUE, and (a 2 , 1), for the (symmetric) JUE, shows that the gap probabilities maybe determined as the product of the smallest eigenvalue distributions of the LUE with parameter α = −1/2, and α = 1/2 and the (shifted) JUE with weights x 1/2 (1 − x) β andThe σ function, namely, the derivative of the log of the smallest eigenvalue distributions of the finite-n LUE or the JUE, satisfies the Jimbo-Miwa-Okamoto σ form of P V and P V I , although in the shift Jacobi case, with the weight x α (1 − x) β , the β parameter does not show up in the equation. We also obtain the asymptotic expansions for the smallest eigenvalue distributions * lvshulin1989@163.com † yangbrookchen@yahoo.co.uk ‡ Corresponding author and faneg@fudan.edu.cn of the Laguerre unitary and Jacobi unitary ensembles after appropriate double scalings, and obtained the constants in the asymptotic expansion of the gap probablities, expressed in term of the Barnes G− function valuated at special point. IntroductionIn the work of Adler and Van Moerbeke [1], the largest eigenvalue distribution of ensembles of n × n random matrices generated by Gaussian, Laguerre and Jacobi weights for general values of the symmetry parameter β, (not to be confused with the β parameter in the Jacobi weight), has been systematically studied, from the perspective of differential operators involving multiple time variables.The gap probabilities that are studied in this paper, the unitary case, denoted by P(a, n), are represented as Fredholm determinant of an integral operator, from the early papers of Mehta, Gaudin and Dyson [19], [20], [22], [31], and [32]. In [28], the gap probability, where a union of disjoint intervals is free of eigenvalues, the integral operator has the sine kernel K(x, y) := sin(x−y) π(x−y) . The (multiple-) gap probability itself was obtained in an expansion in terms of the resolvent of the integral equation. In a tour de force computations, JMMS showed that in the single interval case where (−b, b) is free of eigenvalues, the quantity σ(τ ) := τ d dτ logdet(I − K (−b,b) ), τ = 2b, satisfies a second order non-linear differential equation. Tracy and Widom [36] studied, the finite n version of such problems, namely, the distribution of the smallest eigenvalue in the Laguerre unitary ensembles, and the largest eigenvalue distribution of the Gaussian unitary ensembles starting from the Christoffel-Darboux or Reproducing Kernel, constructed out of the "natural" orthogonal polynomials, namely the Laguerre and Hermite polynomials, respectively. Through a series of differentiation formulas, Tracy and Widom found the finite n version of the differential equations satisfied by the resolvent a...

show abstract

mentioning

confidence: 99%

Asymptotic gap probability distributions of the Gaussian unitary ensembles and Jacobi unitary ensembles

2018

Self Cite

View full text Add to dashboard Cite

show abstract

“…For all n ≥ 1, the coefficient of x n+4 gives us (25). The coefficient of x n+3 gives us l n,2 = 0 , n ≥ 0 .…”

Section: The Symmetric Laguerre-hahn Class Twomentioning

confidence: 99%

The Symmetric Semi-classical Orthogonal Polynomials of Class Two and Some of Their Extensions

Filipuk

Rebocho

2019

Bull. Malays. Math. Sci. Soc.

View full text Add to dashboard Cite

show abstract

“…In [7,29], the second order differential equation for polynomials P n (x) orthogonal with respect to the weight e −x 2 (1 − χ (−a,a) (x)), x ∈ R, a > 0, was obtained.…”

Section: Weights With a Gapmentioning

confidence: 99%

Orthogonal polynomials, asymptotics, and Heun equations

Chen

Filipuk

Zhan

2019

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of "classical" weights multiplied by suitable "deformation factors", usually dependent on a "time variable" t. From ladder operators [12][13][14]30] one finds second order linear ordinary differential equations for associated orthogonal polynomials with coefficients being rational functions. The Painlevé and related functions appear as the residues of these rational functions.We will be interested in the situation when n, the order of the Hankel matrix and also the degree of the polynomials P n (x) orthogonal with respect to the deformed weights, gets large. We show that the second order linear differential equations satisfied by P n (x) are particular cases of Heun equations when n is large. In some sense, monic orthogonal polynomials generated by deformed weights mentioned below are solutions of a variety of Heun equations. Heun equations are of considerable importance in mathematical physics and in the special cases they degenerate to the hypergeometric and confluent hypergeometric equations (see, for instance, [1,23,36]).In this paper we look at three type of weights: the Jacobi type, the Laguerre type and the weights deformed by the indicator function of (a, b) χ (a,b) and the step function θ(x).In particular, we consider the following Jacobi type weights: 1.

show abstract

Continuous and Discrete Painlevé Equations Arising from the Gap Probability Distribution of the Finite $$n$$ n Gaussian Unitary Ensembles

Cited by 8 publications

References 17 publications

Asymptotic gap probability distributions of the Gaussian unitary ensembles and Jacobi unitary ensembles

Asymptotic gap probability distributions of the Gaussian unitary ensembles and Jacobi unitary ensembles

The Symmetric Semi-classical Orthogonal Polynomials of Class Two and Some of Their Extensions

Orthogonal polynomials, asymptotics, and Heun equations

Contact Info

Product

Resources

About