Tracy–Widom Distributions in Critical Unitary Random Matrix Ensembles and the Coupled Painlevé II System

Xu, Shuai‐Xia; Dai, Dao‐Qing

doi:10.1007/s00220-018-3257-y

Cited by 25 publications

(36 citation statements)

References 73 publications

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“…Finally, we highlight that the s-independent term in (2.2) involves an integral with respect to certain Painlevé transcendents. This phenomena was first observed in [7] in the studies of Painlevé II kernel associated with the Hastings-McLeod solution, and is also confirmed in recent work [42] concerning the Painlevé XXXIV kernel. We expect this feature is true for the large s asymptotics associated with any other Painlevé kernels.…”

Section: Large Gap Asymptoticssupporting

confidence: 73%

“…Remark 2.2. In the literature, the asymptotics of Fredholm determinants of other Painlevé kernels have been investigated in [17] for the kernels built in terms of the Painlevé I hierarchy, in [7] for the Painlevé II kernel associated with the Hastings-McLeod solution, and quite recently in [42] for the Painlevé XXXIV kernel. All these Painlevé kernels describe certain critical behaviors encountered in random matrix theory.…”

Section: Large Gap Asymptoticsmentioning

confidence: 99%

See 1 more Smart Citation

Gap Probability at the Hard Edge for Random Matrix Ensembles with Pole Singularities in the Potential

Dai¹,

Xu²,

Zhang³

2018

SIAM J. Math. Anal.

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We study the Fredholm determinant of an integrable operator acting on the interval (0, s) whose kernel is constructed out of the Ψ-function associated with a hierarchy of higher order analogues to the Painlevé III equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probability at the hard edge of unitary invariant random matrix ensembles perturbed by poles of order k in a certain scaling regime. Using the Riemann-Hilbert method, we obtain the large s asymptotics of the Fredholm determinant. Moreover, we derive a Painlevé type formula of the Fredholm determinant, which is expressed in terms of an explicit integral involving a solution to a coupled Painlevé III system.

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Section: Large Gap Asymptoticssupporting

confidence: 73%

Section: Large Gap Asymptoticsmentioning

confidence: 99%

Gap Probability at the Hard Edge for Random Matrix Ensembles with Pole Singularities in the Potential

Dai¹,

Xu²,

Zhang³

2018

SIAM J. Math. Anal.

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“…Simultaneously with our work, a determinant closely related to

F (x_{1}, x_{2}; 0, s)

but with an extra spectral singularity was studied by Xu and Dai in [, theorem 2], where they obtained a Tracy–Widom formula which is equivalent to ours if the spectral singularity is absent.…”

Section: Examples and Applicationsmentioning

confidence: 57%

The Generating Function for the Airy Point Process and a System of Coupled Painlevé II Equations

Claeys

Doeraene

2018

Stud Appl Math

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For a wide class of Hermitian random matrices, the limit distribution of the eigenvalues close to the largest one is governed by the Airy point process. In such ensembles, the limit distribution of the kth largest eigenvalue is given in terms of the Airy kernel Fredholm determinant or in terms of Tracy–Widom formulas involving solutions of the Painlevé II equation. Limit distributions for quantities involving two or more near‐extreme eigenvalues, such as the gap between the kth and the ℓth largest eigenvalue or the sum of the k largest eigenvalues, can be expressed in terms of Fredholm determinants of an Airy kernel with several discontinuities. We establish simple Tracy–Widom type expressions for these Fredholm determinants, which involve solutions to systems of coupled Painlevé II equations, and we investigate the asymptotic behavior of these solutions.

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“…where u(s) = u(s; 2α, ω) is the Painlevé XXXIV transcendent (1.27) and c 0 is the constant given (1.10); see [51,Theorem 4]. As mentioned before, since K P 34 α,0 (x, y; t) describes the transition from the soft edge to the hard edge as t varies from +∞ to −∞, it is expected to observe this interesting transition from the asymptotics (1.48) as well.…”

Section: Phase Transition From the Soft Edge To The Hard Edge -A Viewmentioning

confidence: 99%

On integrals of the tronquée solutions and the associated Hamiltonians for the Painlevé II equation

Dai

Zhang

2020

Journal of Differential Equations

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We consider a family of tronquée solutions of the Painelvé II equationwhich is characterized by the Stokes multiplierswith ω being a free parameter. These solutions include the well-known generalized Hastings-McLeod solution as a special case if ω = 0. We derive asymptotics of integrals of the tronquée solutions and the associated Hamiltonians over the real axis for α > −1/2 and ω ≥ 0, with the constant terms evaluated explicitly. Our results agree with those already known in the literature if the parameters α and ω are chosen to be special values. Some applications of our results in random matrix theory are also discussed.

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Tracy–Widom Distributions in Critical Unitary Random Matrix Ensembles and the Coupled Painlevé II System

Cited by 25 publications

References 73 publications

Gap Probability at the Hard Edge for Random Matrix Ensembles with Pole Singularities in the Potential

Gap Probability at the Hard Edge for Random Matrix Ensembles with Pole Singularities in the Potential

The Generating Function for the Airy Point Process and a System of Coupled Painlevé II Equations

On integrals of the tronquée solutions and the associated Hamiltonians for the Painlevé II equation

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