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On the Characteristic Polynomial of the Eigenvalue Moduli of Random Normal Matrices

Byun, Sung Soo and Charlier, Christophe LU (2024) In Constructive Approximation
Abstract

We study the characteristic polynomial pn(x)=∏j=1n(|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[euπIm\,lnpn(r)eaRe\,lnpn(r)], in the case where r is in the bulk, u ϵ R and a ϵ 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,... (More)

We study the characteristic polynomial pn(x)=∏j=1n(|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[euπIm\,lnpn(r)eaRe\,lnpn(r)], in the case where r is in the bulk, u ϵ R and a ϵ 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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author
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type
Contribution to journal
publication status
epub
subject
keywords
Asymptotic analysis, Jump- and root-type singularities along circles, Moment generating functions, Random matrix theory
in
Constructive Approximation
publisher
Springer
external identifiers
  • scopus:85202067711
ISSN
0176-4276
DOI
10.1007/s00365-024-09689-x
language
English
LU publication?
yes
id
bfceabdb-b3b1-4a6f-bc3b-11be0b1e2d1d
date added to LUP
2024-11-01 09:07:57
date last changed
2025-04-04 15:17:33
@article{bfceabdb-b3b1-4a6f-bc3b-11be0b1e2d1d,
  abstract     = {{<p>We study the characteristic polynomial p<sub>n</sub>(x)=∏<sub>j=1</sub><sup>n</sup>(|z<sub>j</sub>|-x) where the z<sub>j</sub> 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<sup>uπIm\,lnpn(r)eaRe\,lnpn(r)</sup>], in the case where r is in the bulk, u ϵ R and a ϵ 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.</p>}},
  author       = {{Byun, Sung Soo and Charlier, Christophe}},
  issn         = {{0176-4276}},
  keywords     = {{Asymptotic analysis; Jump- and root-type singularities along circles; Moment generating functions; Random matrix theory}},
  language     = {{eng}},
  publisher    = {{Springer}},
  series       = {{Constructive Approximation}},
  title        = {{On the Characteristic Polynomial of the Eigenvalue Moduli of Random Normal Matrices}},
  url          = {{http://dx.doi.org/10.1007/s00365-024-09689-x}},
  doi          = {{10.1007/s00365-024-09689-x}},
  year         = {{2024}},
}