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Exponential moments for disk counting statistics at the hard edge of random normal matrices

Ameur, Yacin LU ; Charlier, Christophe LU ; Cronvall, Joakim LU and Lenells, Jonatan LU (2023) In Journal of Spectral Theory 13(3). p.841-902
Abstract

We consider the multivariate moment generating function of the disk counting statistics of a model Mittag-Leffler ensemble in the presence of a hard wall. Let n be the number of points. We focus on two regimes: (a) the “hard edge regime” where all disk boundaries are at a distance of order n1 from the hard wall, and (b) the “semi-hard edge regime” where all disk boundaries are at a distance of order √1n from the hard wall. As n → + ∞, we prove that the moment generating function enjoys asymptotics of the form (Equation presented) In both cases, we determine the constants C1;:::; C4 explicitly. We also derive precise asymptotic formulas for all joint cumulants of the disk... (More)

We consider the multivariate moment generating function of the disk counting statistics of a model Mittag-Leffler ensemble in the presence of a hard wall. Let n be the number of points. We focus on two regimes: (a) the “hard edge regime” where all disk boundaries are at a distance of order n1 from the hard wall, and (b) the “semi-hard edge regime” where all disk boundaries are at a distance of order √1n from the hard wall. As n → + ∞, we prove that the moment generating function enjoys asymptotics of the form (Equation presented) In both cases, we determine the constants C1;:::; C4 explicitly. We also derive precise asymptotic formulas for all joint cumulants of the disk counting function, and establish several central limit theorems. Surprisingly, and in contrast to the “bulk”, “soft edge”, and “semi-hard edge” regimes, the second and higher order cumulants of the disk counting function in the “hard edge” regime are proportional to n and not to √n.

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organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
asymptotic analysis, Moment generating functions, random matrix theory
in
Journal of Spectral Theory
volume
13
issue
3
pages
62 pages
publisher
European Mathematical Society Publishing House
external identifiers
  • scopus:85178376622
ISSN
1664-039X
DOI
10.4171/JST/474
language
English
LU publication?
yes
id
56c9a98a-2eac-47fc-ae5d-4bf0ab0a8960
date added to LUP
2024-01-04 14:53:24
date last changed
2024-01-04 14:53:51
@article{56c9a98a-2eac-47fc-ae5d-4bf0ab0a8960,
  abstract     = {{<p>We consider the multivariate moment generating function of the disk counting statistics of a model Mittag-Leffler ensemble in the presence of a hard wall. Let n be the number of points. We focus on two regimes: (a) the “hard edge regime” where all disk boundaries are at a distance of order <sub>n</sub><sup>1</sup> from the hard wall, and (b) the “semi-hard edge regime” where all disk boundaries are at a distance of order <sup>√1</sup><sub>n</sub> from the hard wall. As n → + ∞, we prove that the moment generating function enjoys asymptotics of the form (Equation presented) In both cases, we determine the constants C<sub>1</sub>;:::; C<sub>4</sub> explicitly. We also derive precise asymptotic formulas for all joint cumulants of the disk counting function, and establish several central limit theorems. Surprisingly, and in contrast to the “bulk”, “soft edge”, and “semi-hard edge” regimes, the second and higher order cumulants of the disk counting function in the “hard edge” regime are proportional to n and not to √n.</p>}},
  author       = {{Ameur, Yacin and Charlier, Christophe and Cronvall, Joakim and Lenells, Jonatan}},
  issn         = {{1664-039X}},
  keywords     = {{asymptotic analysis; Moment generating functions; random matrix theory}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{841--902}},
  publisher    = {{European Mathematical Society Publishing House}},
  series       = {{Journal of Spectral Theory}},
  title        = {{Exponential moments for disk counting statistics at the hard edge of random normal matrices}},
  url          = {{http://dx.doi.org/10.4171/JST/474}},
  doi          = {{10.4171/JST/474}},
  volume       = {{13}},
  year         = {{2023}},
}