Estimates of the orthogonal polynomials with weight exp(−xm), m an even positive integer

Bonan, Stanford S.; Clark, Dean S.

doi:10.1016/0021-9045(86)90074-2

Cited by 31 publications

(18 citation statements)

References 3 publications

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“…This was already observed for the first time by J. Shohat in 1930 [16]. Considerably later Shohat's idea was developed by Bonan and Clark [17]. The simple and elegant method proposed in Refs.…”

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confidence: 70%

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Novel Universal Correlations in Invariant Random-Matrix Models

Kanzieper¹,

Freilikher²

1997

Phys. Rev. Lett.

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We show that eigenvalue correlations in unitary-invariant ensembles of large random matrices adhere to novel universal laws that only depend on a multicriticality of the bulk density of states near the soft edge of the spectrum. Our consideration is based on the previously unknown observation that genuine density of states and n−point correlation function are completely determined by the Dyson's density analytically continued onto the whole real axis.chao-dyn/9701006

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confidence: 70%

“…The simple and elegant method proposed in Refs. [16,17] turns out to be a very general and powerful one for analysis of spectral properties possessed by large random matrices.…”

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confidence: 99%

Novel Universal Correlations in Invariant Random-Matrix Models

Kanzieper¹,

Freilikher²

1997

Phys. Rev. Lett.

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“…Here we provide a brief guide to the literature on this subject. See, for example, [3,4,10,7,9,8,12,19,20,32,33,38]. In fact, Magnus in [33] noted that such operators were known to Laguerre.…”

Section: Ladder Operators and Non-linear Difference Equationsmentioning

confidence: 95%

Painlevé III and a singular linear statistics in Hermitian random matrix ensembles, I

Chen

Its

2010

Journal of Approximation Theory

132

193

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In this paper, we study a certain linear statistics of the unitary Laguerre ensembles, motivated in part by an integrable quantum field theory at finite temperature. It transpires that this is equivalent to the characterization of a sequence of polynomials orthogonal with respect to the weightnamely, the determination of the associated Hankel determinant and recurrence coefficients. Here w(x, s) is the Laguerre weight x α e −x perturbed by a multiplicative factor e −s/x , which induces an infinitely strong zero at the origin.For polynomials orthogonal on the unit circle, a particular example where there are explicit formulas, the weight of which has infinitely strong zeros, was investigated by Pollaczek and Szegö many years ago. Such weights are said to be singular or irregular due to the violation of the Szegö condition.In our problem, the linear statistics is a sum of the reciprocal of positive random variables {x j : j = 1, . . . , n}; n j=1 1/x j . We show that the moment generating function, or the Laplace transform of the probability density function of this linear statistics, can be expressed as the ratio of Hankel determinants and as an integral involving a particular third Painlevé function.

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“…Bonan, Nevai and others studied orthogonal polynomials with ladder operators. Works on the Hankel determinants generated by the deformations of classical weight, maybe found in [2,8,11,12,13,14], and also in [5,15,18,19,26,27]. Adler and Van Moerbeke obtained differential equations governing the logarithmic derivatives of Hankel determinants, and their generalization, using a multi-time approach, [1].…”

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Singular linear statistics of the Laguerre unitary ensemble and Painlevé. III. Double scaling analysis

Chen

2015

Journal of Mathematical Physics

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We continue with the study of the Hankel determinant, defined by, D n (t, α) = det ∞ 0 x j+k w(x; t, α)dx n−1 j,k=0 , generated by a singularly perturbed Laguerre weight, w(x; t, α) = x α e −x e −t/x , x ∈ R + , α > 0, t > 0, obtained through a deformation of the Laguerre weight function, w(x; 0, α) = x α e −x , x ∈ R + , α > 0, via the multiplicative factor e −t/x . An earlier investigation was made on the finite n aspect of such determinants, which appeared in [20]. It was found that the logarithm of the Hankel determinant has an integral representation in terms of a particular Painlevé III( P III , for short) and its t derivatives. In this paper we show that, under a double scaling, where n , the order of the Hankel matrix tends to ∞, and t , tends to 0 + , the scaled-and therefore, in some sense, infinite dimensional-Hankel determinant, has an integral representation in terms of a C potential. The second order non-linear ode satisfied by C, after a change of variable, is another P III transcendent, albeit with fewer number of parameters. Expansions of the double scaled determinant for small and large parameter are oband Telephone:(853)8822-8546.

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Estimates of the orthogonal polynomials with weight exp(−xm), m an even positive integer

Cited by 31 publications

References 3 publications

Novel Universal Correlations in Invariant Random-Matrix Models

Novel Universal Correlations in Invariant Random-Matrix Models

Painlevé III and a singular linear statistics in Hermitian random matrix ensembles, I

Singular linear statistics of the Laguerre unitary ensemble and Painlevé. III. Double scaling analysis

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