Asymptotics of Hankel Determinants With a One-Cut Regular Potential and Fisher–Hartwig Singularities

Charlier, Christophe

doi:10.1093/imrn/rny009

Cited by 49 publications

(52 citation statements)

References 112 publications

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“…(1.12) Remark 1. The above asymptotics have similarities with the asymptotics for Hankel determinants with m Fisher-Hartwig singularities studied in [15]. This is quite natural, since the Fredholm determinants E( x; β) and E 0 ( x; β 0 ) can be obtained as scaling limits of such Hankel determinants.…”

Section: Introductionsupporting

confidence: 63%

“…In this section, we deal with the case s 1 = 0. The general strategy in this section has many similarities with the analysis in [15], needed in the study of Hankel determinants with several Fisher-Hartwig singularities.…”

Section: Asymptotic Analysis Of Rh Problem For ψ With S 1 =mentioning

confidence: 99%

“…This type of construction is well understood and relies on the solution Ψ HG (z) to a model RH problem, which we recall in Appendix A.3 for the convenience of the reader. For more details about it, we refer to [29,25,15]. Let us first consider the function…”

Section: Local Parametrices Aroundmentioning

confidence: 99%

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Large Gap Asymptotics for Airy Kernel Determinants with Discontinuities

Charlier

Claeys

2019

Commun. Math. Phys.

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We obtain large gap asymptotics for Airy kernel Fredholm determinants with any number m of discontinuities. These m-point determinants are generating functions for the Airy point process and encode probabilistic information about eigenvalues near soft edges in random matrix ensembles. Our main result is that the m-point determinants can be expressed asymptotically as the product of m 1-point determinants, multiplied by an explicit constant pre-factor which can be interpreted in terms of the covariance of the counting function of the process.1 24 e ζ ′ (−1) of the multiplicative constant. For m = 1 with s > 0, it is notationally convenient to write s = e −2πiβ with β ∈ iR, and it was proved only recently by Bothner and Buckingham [13] thatwhere G is Barnes' G-function, confirming a conjecture from [8]. The error term in (1.5) is uniform for β in compact subsets of the imaginary line.We generalize these asymptotics to general values of m, for s 2 , . . . , s m ∈ (0, 1], and s 1 ∈ [0, 1], and show that they exhibit an elegant multiplicative structure. To see this, we need to make a change of variables s → β, by defining β j ∈ iR as follows. If s 1 > 0, we define β = (β 1 , . . . , β m ) bys m for j = m, (1.6) and if s 1 = 0, we define β 0 = (β 2 , . . . , β m ) with β 2 , . . . , β m again defined by (1.6). We then denote, if s 1 > 0, E( x; β) := E   m j=1 e −2πiβjN (x j ,+∞)

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Section: Introductionsupporting

confidence: 63%

Section: Asymptotic Analysis Of Rh Problem For ψ With S 1 =mentioning

confidence: 99%

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Large Gap Asymptotics for Airy Kernel Determinants with Discontinuities

Charlier

Claeys

2019

Commun. Math. Phys.

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“…In the regime when s is in a compact subset of (0, 1] and v = √ 2nt with t in a compact subset of (−1, 1), asymptotics of this probability have been obtained rigorously in [27] for α = 0 and in [9] for α = 0. Note that asymptotics for integer α were obtained previously in [25], based on the work [7].…”

Section: Introductionmentioning

confidence: 99%

Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity

Charlier¹,

Deaño²

2018

SIGMA

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We study n × n Hankel determinants constructed with moments of a Hermite weight with a Fisher-Hartwig singularity on the real line. We consider the case when the singularity is in the bulk and is both of root-type and jump-type. We obtain large n asymptotics for these Hankel determinants, and we observe a critical transition when the size of the jumps varies with n. These determinants arise in the thinning of the generalised Gaussian unitary ensembles and in the construction of special function solutions of the Painlevé IV equation.

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“…The Riemann-Hilbert method [18] is a very powerful tool to investigate the asymptotic behavior of many unitary ensembles. See, for instance, the gap probability problem [16,42], correlation kernel [10,40], partition functions [2,8,17], Hankel determinants and orthogonal polynomials [7,9,11,41]. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

The largest eigenvalue distribution of the laguerre unitary ensemble

Lyu

Chen

2017

Acta Mathematica Scientia

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We study the probability that all the eigenvalues of n × n Hermitian matrices, from the Laguerre unitary ensemble with the weight x γ e −4nx , x ∈ [0, ∞), γ > −1, lie in the interval [0, α]. By using previous results for finite n obtained by the ladder operator approach of orthogonal polynomials, we derive the large n asymptotics of the largest eigenvalue distribution function with α ranging from 0 to the soft edge. In addition, at the soft edge, we compute the constant conjectured by Tracy and Widom [Commun. Math. Phys. 159 (1994), 151-174], later proved by Deift, Its and Krasovsky [Commun. Math. Phys. 278 (2008), 643-678]. Our results are reduced to those of Deift et al. when γ = 0.

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Asymptotics of Hankel Determinants With a One-Cut Regular Potential and Fisher–Hartwig Singularities

Cited by 49 publications

References 112 publications

Large Gap Asymptotics for Airy Kernel Determinants with Discontinuities

Large Gap Asymptotics for Airy Kernel Determinants with Discontinuities

Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity

The largest eigenvalue distribution of the laguerre unitary ensemble

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