Orthogonal and Multiple Orthogonal Polynomials, Random Matrices, and Painlevé Equations

Assche, Walter Van

doi:10.1007/978-3-030-36744-2_22

Cited by 18 publications

(28 citation statements)

References 44 publications

(40 reference statements)

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“…The relevance of Painlevé transcendents in random matrix theory arose most famously in the work of Tracy and Widom in connection with the largest eigenvalue of a Hermitian random matrix [63], building on the earlier developments of Jimbo, Miwa, Môri and Sato [52]. They are now among the most important special functions of mathematical physics, and they appear in connection with reduction of integrable PDEs, models in statistical mechanics, combinatorics and orthogonal polynomials, to name only a few applications, see [65] for a recent survey, and the manuscript [34]. The characterization of averages involving Hermitian (or unitary) random matrices in terms of Painlevé transcendents has attracted considerable activity in the last decades.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Characteristic Polynomials of Complex Random Matrices and Painlevé Transcendents

Deaño

Simm

2020

International Mathematics Research Notices

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We study expectations of powers and correlation functions for characteristic polynomials of $N \times N$ non-Hermitian random matrices. For the $1$-point and $2$-point correlation function, we obtain several characterizations in terms of Painlevé transcendents, both at finite $N$ and asymptotically as $N \to \infty $. In the asymptotic analysis, two regimes of interest are distinguished: boundary asymptotics where parameters of the correlation function can touch the boundary of the limiting eigenvalue support and bulk asymptotics where they are strictly inside the support. For the complex Ginibre ensemble this involves Painlevé IV at the boundary as $N \to \infty $. Our approach, together with the results in [ 49], suggests that this should arise in a much broader class of planar models. For the bulk asymptotics, one of our results can be interpreted as the merging of two “planar Fisher–Hartwig singularities” where Painlevé V arises in the asymptotics. We also discuss the correspondence of our results with a normal matrix model with $d$-fold rotational symmetries known as the lemniscate ensemble, recently studied in [ 15, 18]. Our approach is flexible enough to apply to non-Gaussian models such as the truncated unitary ensemble or induced Ginibre ensemble; we show that in the former case Painlevé VI arises at finite $N$. Scaling near the boundary leads to Painlevé V, in contrast to the Ginibre ensemble.

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Section: Introduction and Resultsmentioning

confidence: 99%

Characteristic Polynomials of Complex Random Matrices and Painlevé Transcendents

Deaño

Simm

2020

International Mathematics Research Notices

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“…For further information about the discrete Painlevé I hierarchy, see [24,25]. Equations (4.17) and (4.18) with t=0 were derived by Freud [18]; see also [5,14]. Further, equations (4.17) and (4.18) with λ=−false012 are also known as ‘string equations’ and arise in important physical applications such as two-dimensional quantum gravity, cf.…”

Section: Recurrence Coefficients For Generalized Higher-order Freud W...mentioning

confidence: 99%

Generalized higher-order Freud weights

2023

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We discuss polynomials orthogonal with respect to a semi-classical generalized higher-order Freud weight ω ( x ; t , λ ) = | x | 2 λ + 1 exp ⁡ ( t x 2 − x 2 m ) , x ∈ R , with parameters λ > − 1 , t ∈ R and m = 2 , 3 , … . The sequence of generalized higher-order Freud weights for m = 2 , 3 , … , forms a hierarchy of weights, with associated hierarchies for the first moment and the recurrence coefficient. We prove that the first moment can be written as a finite partition sum of generalized hypergeometric 1 F m functions and show that the recurrence coefficients satisfy difference equations which are members of the first discrete Painlevé hierarchy. We analyse the asymptotic behaviour of the recurrence coefficients and the limiting distribution of the zeros as n → ∞ . We also investigate structure and other mixed recurrence relations satisfied by the polynomials and related properties.

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“…as z → 0, where |⃗ n| = p j=0 n j . For more information on Hermite-Padé approximation and multiple orthogonal polynomials, we refer to [19] and references therein.…”

Section: Type I Multiple Orthogonalitymentioning

confidence: 99%

anar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials

Berezin¹,

Kuijlaars²,

Parra³

2023

SIGMA

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A recent result of S.-Y. Lee and M. Yang states that the planar orthogonal polynomials orthogonal with respect to a modified Gaussian measure are multiple orthogonal polynomials of type II on a contour in the complex plane. We show that the same polynomials are also type I orthogonal polynomials on a contour, provided the exponents in the weight are integer. From this orthogonality, we derive several equivalent Riemann-Hilbert problems. The proof is based on the fundamental identity of Lee and Yang, which we establish using a new technique.

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Orthogonal and Multiple Orthogonal Polynomials, Random Matrices, and Painlevé Equations

Cited by 18 publications

References 44 publications

Characteristic Polynomials of Complex Random Matrices and Painlevé Transcendents

Characteristic Polynomials of Complex Random Matrices and Painlevé Transcendents

Generalized higher-order Freud weights

anar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials

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