Asymptotics of a Fredholm Determinant Corresponding to the First Bulk Critical Universality Class in Random Matrix Models

Bothner, Thomas; Its, Alexander

doi:10.1007/s00220-014-1950-z

Cited by 13 publications

(13 citation statements)

References 44 publications

(65 reference statements)

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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: 72%

“…[1]. Combining (7.25), (8.5) and the above formula, we obtain 7) which is given in (2.11). This completes the proofs of Theorem 2.4 and (2.11).…”

Section: Asymptotic Analysis Of the Riemann-hilbert Problem For Mmentioning

confidence: 99%

“…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 2 more Smart Citations

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: 72%

“…[1]. Combining (7.25), (8.5) and the above formula, we obtain 7) which is given in (2.11). This completes the proofs of Theorem 2.4 and (2.11).…”

Section: Asymptotic Analysis Of the Riemann-hilbert Problem For Mmentioning

confidence: 99%

Section: Large Gap Asymptoticsmentioning

confidence: 99%

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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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“…REMARK 7. For α = 0, the above large gap asymptotics can be reduced to that in Bothner and Its [12] ln det…”

Section: )mentioning

confidence: 97%

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

Dai

2018

Commun. Math. Phys.

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We study Fredholm determinants of the Painlevé II and Painlevé XXXIV kernels. In certain critical unitary random matrix ensembles, these determinants describe special gap probabilities of eigenvalues. We obtain Tracy-Widom formulas for the Fredholm determinants, which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled Painlevé II system in dimension four. Moreover, the large gap asymptotics for these Fredholm determinants are derived, where the constant terms are given explicitly in terms of the Riemann zeta-function.

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“…This problem appears not only in exactly solvable models of statistical mechanics, such as the six-vertex model and the Ising model, but also in random matrix theory, combinatorics, theory of integrable PDEs, etc. In different settings, the "constant factor problem" is studied in the works of Tracy [26], Basor and Tracy [3], Budylin and Buslaev [5], Ehrhardt [19], Deift, Its, Krasovsky, and Zhou [14], Deift, Its, and Krasovsky [15], Baik, Buckingham, and DiFranco [2], Bothner and Its [11], Forrester [22], and others.…”

Section: 18)mentioning

confidence: 99%

Calculation of the constant factor in the six-vertex model

Bleher¹,

Bothner²

2014

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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We calculate explicitly the constant factor C in the large N asymptotics of the partition function ZN of the six-vertex model with domain wall boundary conditions on the critical line between the disordered and ferroelectric phases. On the critical line the weights a, b, c of the model are parameterized by a parameter α > 1, as a = α−1 2 , b = α+1 2 , c = 1. The asymptotics of ZN on the critical line was obtained earlier in the paper [8] of Bleher and Liechty:where F and G are given by explicit expressions, but the constant factor C > 0 was not known. To calculate the constant C, we find, by using the Riemann-Hilbert approach, an asymptotic behavior of ZN in the double scaling limit, as N and α tend simultaneously to ∞ in such a way that N α → t ≥ 0. Then we apply the Toda equation for the tau-function to find a structural form for C, as a function of α, and we combine the structural form of C and the double scaling asymptotic behavior of ZN to calculate C.

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Asymptotics of a Fredholm Determinant Corresponding to the First Bulk Critical Universality Class in Random Matrix Models

Cited by 13 publications

References 44 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

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

Calculation of the constant factor in the six-vertex model

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