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Dieng, Momar

doi:10.1155/imrn.2005.2263

Cited by 29 publications

(10 citation statements)

References 16 publications

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“…which shows that X(z) in ( 110) satisfies (25) and in turn (27) as z → ∞ along any non-horizontal direction.…”

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confidence: 85%

A Riemann-Hilbert approach to Fredholm determinants of Hankel composition operators: scalar-valued kernels

Bothner¹

2022

Preprint

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We characterize Fredholm determinants of a class of Hankel composition operators via matrixvalued Riemann-Hilbert problems, for additive and multiplicative compositions. The scalar-valued kernels of the underlying integral operators are not assumed to display the integrable structure known from the seminal work of Its, Izergin, Korepin and Slavnov [41]. Yet we are able to describe the corresponding Fredholm determinants through a naturally associated Riemann-Hilbert problem of Zakharov-Shabat type by solely exploiting the kernels' Hankel composition structures. We showcase the efficiency of this approach through a series of examples, we then compute several rank one perturbed determinants in terms of Riemann-Hilbert data and finally derive Akhiezer-Kac asymptotic theorems for suitable kernel classes.

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“…which shows that X(z) in ( 110) satisfies (25) and in turn (27) as z → ∞ along any non-horizontal direction.…”

mentioning

confidence: 85%

A Riemann-Hilbert approach to Fredholm determinants of Hankel composition operators: scalar-valued kernels

Bothner¹

2022

Preprint

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“…Identities similar to (1.10) have been derived in [13,Proposition 1.1] for the limiting GOE and the limiting Gaussian symplectic ensemble (GSE) based on Painlevé representations for the underlying eigenvalue generating functions, cf. [18,Theorem 2.1]. Our proof of Lemma 1.7 will rely on the observation that thinned Pfaffian point processes are Pfaffian with an appropriately γ-modified kernel, see Section 2 below, which is similar to the proof for determinantal point processes given in [38,Appendix A].…”

Section: Riemann-hilbertmentioning

confidence: 99%

“…Afterwards we use (1.10) and carefully simplify the regularized Fredholm determinant in order to arrive at a finite n formula which is amenable to asymptotics. Our approach is somewhat similar to the ones carried out in [18,42], however two issues arise along the way: one, the absence of Christoffel-Darboux structures throughout forces us to rely on the Fourier tricks used in [2, Section 2 and 3] in the derivation of (1.14). Two, unlike in the invariant ensembles, our computations depend heavily on the parity of n. We first work out the necessary details for even n in Section 3 and afterwards develop a comparison argument to treat all odd n, see Subsection 3.3.…”

Section: Riemann-hilbertmentioning

confidence: 99%

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Edge distribution of thinned real eigenvalues in the real Ginibre ensemble

Baik¹,

Bothner²

2020

Preprint

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This paper is concerned with the explicit computation of the limiting distribution function of the largest real eigenvalue in the real Ginibre ensemble when each real eigenvalue has been removed independently with constant likelihood. We show that the recently discovered integrable structures in [2] generalize from the real Ginibre ensemble to its thinned equivalent. Concretely, we express the aforementioned limiting distribution function as a convex combination of two simple Fredholm determinants and connect the same function to the inverse scattering theory of the Zakharov-Shabat system. As corollaries, we provide a Zakharov-Shabat evaluation of the ensemble's real eigenvalue generating function and obtain precise control over the limiting distribution function's tails. The latter part includes the explicit computation of the usually difficult constant factors.

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“…We remark that the above expression for the GSE case was obtained in Refs. [19,20] using a scaling that assume N/2 eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

Deformations of the Tracy-Widom distribution

2009

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In random matrix theory, the Tracy-Widom (TW) distribution describes the behavior of the largest eigenvalue. We consider here two models in which TW undergoes transformations. In the first one disorder is introduced in the Gaussian ensembles by superimposing an external source of randomness. A competition between TW and a normal (Gaussian) distribution results, depending on the spreading of the disorder. The second model consists of removing at random a fraction of (correlated) eigenvalues of a random matrix. The usual formalism of Fredholm determinants extends naturally. A continuous transition from TW to the Weilbull distribution, characteristic of extreme values of an uncorrelated sequence, is obtained.

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Cited by 29 publications

References 16 publications

A Riemann-Hilbert approach to Fredholm determinants of Hankel composition operators: scalar-valued kernels

A Riemann-Hilbert approach to Fredholm determinants of Hankel composition operators: scalar-valued kernels

Edge distribution of thinned real eigenvalues in the real Ginibre ensemble

Deformations of the Tracy-Widom distribution

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