Asymptotics of Hankel determinants with a Laguerre-type or Jacobi-type potential and Fisher-Hartwig singularities

Charlier, Christophe; Gharakhloo, Roozbeh

doi:10.48550/arxiv.1902.08162

Cited by 7 publications

(23 citation statements)

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“…α ∼ µ. For λ = 0 our result (48) agrees with the Fisher-Hartwig asymptotics obtained using Riemann Hilbert methods for the LUE in [73], which assumes α = O(1) (see also Section X A). Finally, one can check that as ã − b 1, Eq.…”

Section: More Details On the Results For The Counting Statistics In D =supporting

confidence: 85%

“…Using (44) and trigonometric relations, one obtains explicit formula (not displayed here) for VarN [a,b] from (2) for a, b in the bulk, and for VarN [0,a] = VarN [a,L] from (16), in the regime a ∼ b ∼ µ. One can check that in the limit a ∼ b µ they agree with the Fisher-Hartwig asymptotics obtained using Riemann-Hilbert methods for the JUE in [73] (see also Section X A). Here we only display the result for the hard box V (x) = 0 for x ∈ [0, L] and Dirichlet boundary conditions for the wavefunction at x = 0, L. It can be obtained as the limit of the JUE for a = b = 1/2, i.e.…”

Section: More Details On the Results For The Counting Statistics In D =supporting

confidence: 61%

“…One can ask about higher cumulants of N D . In d = 1, for the HO and the inverse square well, they can be extracted from known Fisher-Hartwig asymptotics of Hankel and Toeplitz determinants [19,53,73,80]. In all cases we find for n…”

Section: Var(mentioning

confidence: 91%

“…In d = 1, for the specific potentials and geometries related to RMT described in Section II the higher cumulants of the number of fermions N I in any interval I can be extracted from known results in random matrix theory, using the mapping between the fermion positions x i and the eigenvalues λ i . In RMT the generating function of the cumulants is computed from Fisher-Hartwig type asymptotics of Hankel and Toeplitz determinants, using Riemann Hilbert methods [19,20,53,73,80]. Let us focus on Ref.…”

Section: A Higher Cumulantsmentioning

confidence: 99%

See 3 more Smart Citations

Counting statistics for non-interacting fermions in a $d$-dimensional potential

Smith,

Doussal,

Majumdar

et al. 2020

Preprint

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Section: More Details On the Results For The Counting Statistics In D =supporting

confidence: 85%

Section: More Details On the Results For The Counting Statistics In D =supporting

confidence: 61%

Section: Var(mentioning

confidence: 91%

Section: A Higher Cumulantsmentioning

confidence: 99%

See 2 more Smart Citations

Counting statistics for non-interacting fermions in a $d$-dimensional potential

Smith,

Doussal,

Majumdar

et al. 2020

Preprint

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“…We also refer to the monographs [8] and [9] as the main sources for general facts concerning the theory of Toeplitz matrices and operators. The asymptotic results concerning the Hankel determinants and their applications -both recent and classical, are featured in the papers [15], [10], [11], [20], and [19] and in the references therein.…”

Section: Introductionmentioning

confidence: 99%

A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants

Gharakhloo,

Its

2019

Preprint

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In this paper we will formulate 4 × 4 Riemann-Hilbert problems for Toeplitz+Hankel determinants and the associated system of orthogonal polynomials, when the Hankel symbol is supported on the unit circle and also when it is supported on an interval [a, b], 0 < a < b < 1. The distinguishing feature of this work is that in the formulation of the Riemann-Hilbert problem no specific relationship is assumed between the Toeplitz and Hankel symbols. We will develop nonlinear steepest descent methods for analysing these problems in the case where the symbols are smooth (i.e. in the absence of Fisher-Hartwig singularities) and admit an analytic continuation in a neighborhood of the unit circle (if the symbol's support is the unit circle). We will finally introduce a model problem and will present its solution requiring certain conditions on the ratio of Hankel and Toeplitz symbols. This in turn will allow us to find the asymptotics of the norms h n of the corresponding orthogonal polynomials and, in fact, the large n asymptotics of the polynomials themselves. We will explain how this solvable case is related to the recent operator-theoretic approach in [6] to Toeplitz+Hankel determinants. At the end we will discuss the prospects of future work and outline several technical, as well as conceptual, issues which we are going to address next within the 4 × 4 Riemann-Hilbert framework introduced in this paper.Notation. Throughout the paper we will frequently use the notation f (z), to denote f (z −1 ).

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Painlevé V and the Hankel determinant for a singularly perturbed Jacobi weight

Min

Chen

2020

Nuclear Physics B

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We study the Hankel determinant generated by a singularly perturbed Jacobi weightIf t = 0, it is reduced to the classical symmetric Jacobi weight. For t > 0, the factor e − t x 2 induces an infinitely strong zero at the origin. This Hankel determinant is related to the Wigner time-delay distribution in chaotic cavities.In the finite n dimensional case, we obtain two auxiliary quantities R n (t) and r n (t) by using the ladder operator approach. We show that the Hankel determinant has an integral representation in terms of R n (t), where R n (t) is closely related to a particular Painlevé V transcendent. Furthermore, we derive a second-order nonlinear differential equation and also a second-order difference equation for the logarithmic derivative of the Hankel determinant.This quantity can be expressed in terms of the Jimbo-Miwa-Okamoto σ-function of a particular Painlevé V. Then we consider the asymptotics of the Hankel determinant under a suitable double scaling, i.e. n → ∞ and t → 0 such that s = 2n 2 t is fixed. Based on previous results by using the Coulomb fluid method, we obtain the large s and small s asymptotic behaviors of the scaled Hankel determinant, including the constant term in the asymptotic expansion.

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Asymptotics of Hankel determinants with a Laguerre-type or Jacobi-type potential and Fisher-Hartwig singularities

Cited by 7 publications

References 0 publications

Counting statistics for non-interacting fermions in a $d$-dimensional potential

Counting statistics for non-interacting fermions in a $d$-dimensional potential

A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants

Painlevé V and the Hankel determinant for a singularly perturbed Jacobi weight

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