Asymptotics of Muttalib–Borodin determinants with Fisher–Hartwig singularities

Charlier, Christophe

doi:10.1007/s00029-022-00762-6

Cited by 7 publications

(5 citation statements)

References 46 publications

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“…There exists also a vast literature on other structured determinants with Fisher-Hartwig singularities, see e.g. [25,26] for Fredholm determinants, [7,35,23] for Toeplitz+Hankel determinants, and [19] for a biorthogonal generalization of Hankel determinants.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On the characteristic polynomial of the eigenvalue moduli of random normal matrices

Byun¹,

Charlier²

2022

Preprint

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We study the characteristic polynomial pn(x) = n j=1 (|zj| − x) where the zj are drawn from the Mittag-Leffler ensemble, i.e. a two-dimensional determinantal point process which generalizes the Ginibre point process. We obtain precise large n asymptotics for the moment generating function E[e u π Im ln pn(r) e a Re ln pn (r) ], in the case where r is in the bulk, u ∈ R and a ∈ N. This expectation involves an n × n determinant whose weight is supported on the whole complex plane, is rotation-invariant, and has both jump-and root-type singularities along the circle centered at 0 of radius r. This "circular" root-type singularity differs from earlier works on Fisher-Hartwig singularities, and surprisingly yields a new kind of ingredient in the asymptotics, the so-called associated Hermite polynomials.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On the characteristic polynomial of the eigenvalue moduli of random normal matrices

Byun¹,

Charlier²

2022

Preprint

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“…In this regard, the formulation of a (3 × 3) Riemann-Hilbert problem corresponding to these bi-orthogonal polynomial systems, both as a means of founding the whole theory upon this and deriving key results but also to pave the way for a suitable Deift-Zhou analysis will be the topic of a future publication. At a later stage, the asymptotic description of these determinants can be investigated when the symbol w is of Fisher-Hartwig type, similar to what has been done for Toeplitz [26], Hankel [18,20], and Muttalib-Borodin [19] determinants.…”

Section: Prospects Of Future Workmentioning

confidence: 99%

Modulated Bi-Orthogonal Polynomials on the Unit Circle: The $$2j-k$$ and $$j-2k$$ Systems

Gharakhloo

Witte

2023

Constr Approx

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We construct the systems of bi-orthogonal polynomials on the unit circle where the Toeplitz structure of the moment determinants is replaced by $$\det (w_{2j-k})_{0\le j,k \le N-1} $$ det ( w 2 j - k ) 0 ≤ j , k ≤ N - 1 and the corresponding Vandermonde modulus squared is replaced by $$\prod _{1 \le j < k \le N}(\zeta _k - \zeta _j)(\zeta ^{-2}_k - \zeta ^{-2}_j) $$ ∏ 1 ≤ j < k ≤ N ( ζ k - ζ j ) ( ζ k - 2 - ζ j - 2 ) . This is the simplest case of a general system of $$pj-qk$$ p j - q k with p, q co-prime integers. We derive analogues of the structures well known in the Toeplitz case: third order recurrence relations, determinantal and multiple-integral representations, their reproducing kernel and Christoffel–Darboux sum, and associated (Carathéodory) functions. We close by giving full explicit details for the system defined by the simple weight $$ w(\zeta )=e^{\zeta }$$ w ( ζ ) = e ζ , which is a specialisation of a weight arising from averages of moments of derivatives of characteristic polynomials over $$\textrm{USp}(2N)$$ USp ( 2 N ) , $$\textrm{SO}(2N)$$ SO ( 2 N ) and $$\textrm{O}^-(2N)$$ O - ( 2 N ) .

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“…By combining lemmas 5.1 and 5.2, we arrive at the following result (the proof is very similar to [15,Proof of (1.38)], so we omit it). Lemma 5.3.…”

Section: 6)mentioning

confidence: 83%

“…Proof. A naive adaptation of [15,Lemma 8.1] (an important difference between [15] and our situation is that σ n = √ log n √ 2π in [15] while here we have…”

Section: 6)mentioning

confidence: 96%

Toeplitz determinants with a one-cut regular potential and Fisher–Hartwig singularities I. Equilibrium measure supported on the unit circle

Blackstone¹,

Charlier²,

Lenells³

2023

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We consider Toeplitz determinants whose symbol has: (i) a one-cut regular potential $V$ , (ii) Fisher–Hartwig singularities and (iii) a smooth function in the background. The potential $V$ is associated with an equilibrium measure that is assumed to be supported on the whole unit circle. For constant potentials $V$ , the equilibrium measure is the uniform measure on the unit circle and our formulas reduce to well-known results for Toeplitz determinants with Fisher–Hartwig singularities. For non-constant $V$ , our results appear to be new even in the case of no Fisher–Hartwig singularities. As applications of our results, we derive various statistical properties of a determinantal point process which generalizes the circular unitary ensemble.

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Asymptotics of Muttalib–Borodin determinants with Fisher–Hartwig singularities

Cited by 7 publications

References 46 publications

On the characteristic polynomial of the eigenvalue moduli of random normal matrices

On the characteristic polynomial of the eigenvalue moduli of random normal matrices

Modulated Bi-Orthogonal Polynomials on the Unit Circle: The $$2j-k$$ and $$j-2k$$ Systems

Toeplitz determinants with a one-cut regular potential and Fisher–Hartwig singularities I. Equilibrium measure supported on the unit circle

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