A Matrix Model with a Singular Weight and Painlevé III

Brightmore, L.; Mezzadri, Francesco; Mo, M. Y.

doi:10.1007/s00220-014-2076-z

Cited by 33 publications

(42 citation statements)

References 36 publications

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“…They also obtained asymptotics of the partition function Z n , which is characterized by a solution of a PIII equation. Although the PIII equation in [3] is not the same as that in (1.20) below, similar phase transitions are observed; c.f. Theorems 1-3.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 64%

“…As pointed out in [3,31], when α = ± 1 2 and s = 0, the system of polynomials orthogonal with respect to (1.5) and (1.11) can be mapped to each other by a change of variables, however, the respective partition functions are still different. In [3], Brightmore et al showed that a phase transition emerges as the matrix size n → ∞ and s, z = O(1/ √ n). They also obtained asymptotics of the partition function Z n , which is characterized by a solution of a PIII equation.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…The partition function, i.e., the quantity defined in (1.2), serves as the moment generating function of the probability density of the Wigner delay time [3,32].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…[22,23,43]). The first attempt to study asymptotics of matrix models with an essential singularity was done by Mezzadri and Mo [31] and Brightmore et al [3] when they are considering asymptotic properties of the partition function (the normalization constant Z n (1.2), in our notation) associated with the following weight w(x; z, s) = exp − z 2 2x 2 + s x − x 2 2 , z ∈ R \ {0}, 0 ≤ s < ∞, x ∈ R.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…We think this may reflect a new class of universality under the effect of the essential singularity. It is worth mentioning that, as in the current paper, the Deift-Zhou nonlinear steepest descent method is also used as one of the main tools in [3]. The interested reader may compare different model RH problems in Section 2 and [3, Sec.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

See 4 more Smart Citations

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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Section: Introduction and Statement Of Resultsmentioning

confidence: 64%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…The partition function, i.e., the quantity defined in (1.2), serves as the moment generating function of the probability density of the Wigner delay time [3,32].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

See 3 more Smart Citations

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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Gaussian Unitary Ensembles with Pole Singularities Near the Soft Edge and a System of Coupled Painlevé XXXIV Equations

Dai

Zhang

2019

Ann. Henri Poincaré

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In this paper, we study the singularly perturbed Gaussian unitary ensembles defined by the measure, in the multiple scaling limit where λ → 1 together with t → 0 as n → ∞ at appropriate related rates. We obtain the asymptotics of the partition function, which is described explicitly in terms of an integral involving a smooth solution to a new coupled Painlevé system generalizing the Painlevé XXXIV equation. The large n limit of the correlation kernel is also derived, which leads to a new universal class built out of the Ψ-function associated with the coupled Painlevé system.

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Moments of Random Matrices and Hypergeometric Orthogonal Polynomials

Cunden

Mezzadri

O’Connell

et al. 2019

Commun. Math. Phys.

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We establish a new connection between moments of n × n random matrices Xn and hypergeometric orthogonal polynomials. Specifically, we consider moments E Tr X −s n as a function of the complex variable s ∈ C, whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and orthogonality relations. An application of the theory resolves part of an integrality conjecture of Cunden et al. [F. D. Cunden, F. Mezzadri, N. J. Simm and P. Vivo, J. Math. Phys. 57 (2016)] on the time-delay matrix of chaotic cavities. In each of the classical ensembles of random matrix theory (Gaussian, Laguerre, Jacobi) we characterise the moments in terms of the Askey scheme of hypergeometric orthogonal polynomials. We also calculate the leading order n → ∞ asymptotics of the moments and discuss their symmetries and zeroes. We discuss aspects of these phenomena beyond the random matrix setting, including the Mellin transform of products and Wronskians of pairs of classical orthogonal polynomials. When the random matrix model has orthogonal or symplectic symmetry, we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials. Contents 48References 50 1 arXiv:1805.08760v5 [math-ph]

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A Matrix Model with a Singular Weight and Painlevé III

Abstract: We investigate the matrix model with weight w(x) := exp − z 2 2x 2 + * The authors acknowledge financial support by the EPSRC grant EP/G019843/1.

Cited by 33 publications

References 36 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

Gaussian Unitary Ensembles with Pole Singularities Near the Soft Edge and a System of Coupled Painlevé XXXIV Equations

Moments of Random Matrices and Hypergeometric Orthogonal Polynomials

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