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Disk counting statistics near hard edges of random normal matrices : The multi-component regime

Ameur, Yacin LU ; Charlier, Christophe LU ; Cronvall, Joakim LU and Lenells, Jonatan LU (2024) In Advances in Mathematics 441.
Abstract

We consider a two-dimensional point process whose points are separated into two disjoint components by a hard wall, and study the multivariate moment generating function of the corresponding disk counting statistics. We investigate the “hard edge regime” where all disk boundaries are a distance of order [Formula presented] away from the hard wall, where n is the number of points. We prove that as n→+∞, the asymptotics of the moment generating function are of the form [Formula presented] and we determine the constants C1,…,C4 explicitly. The oscillatory term Fn is of order 1 and is given in terms of the Jacobi theta function. Our theorem allows us to derive various precise results on the disk counting... (More)

We consider a two-dimensional point process whose points are separated into two disjoint components by a hard wall, and study the multivariate moment generating function of the corresponding disk counting statistics. We investigate the “hard edge regime” where all disk boundaries are a distance of order [Formula presented] away from the hard wall, where n is the number of points. We prove that as n→+∞, the asymptotics of the moment generating function are of the form [Formula presented] and we determine the constants C1,…,C4 explicitly. The oscillatory term Fn is of order 1 and is given in terms of the Jacobi theta function. Our theorem allows us to derive various precise results on the disk counting function. For example, we prove that the asymptotic fluctuations of the number of points in one component are of order 1 and are given by an oscillatory discrete Gaussian. Furthermore, the variance of this random variable enjoys asymptotics described by the Weierstrass ℘-function.

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organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Moment generating functions, Oscillatory asymptotics, Random matrix theory
in
Advances in Mathematics
volume
441
article number
109549
publisher
Elsevier
external identifiers
  • scopus:85186094553
ISSN
0001-8708
DOI
10.1016/j.aim.2024.109549
language
English
LU publication?
yes
id
d2ac2cb2-c477-4e80-8998-e8b13e76188e
date added to LUP
2024-03-14 14:42:12
date last changed
2024-03-14 14:42:22
@article{d2ac2cb2-c477-4e80-8998-e8b13e76188e,
  abstract     = {{<p>We consider a two-dimensional point process whose points are separated into two disjoint components by a hard wall, and study the multivariate moment generating function of the corresponding disk counting statistics. We investigate the “hard edge regime” where all disk boundaries are a distance of order [Formula presented] away from the hard wall, where n is the number of points. We prove that as n→+∞, the asymptotics of the moment generating function are of the form [Formula presented] and we determine the constants C<sub>1</sub>,…,C<sub>4</sub> explicitly. The oscillatory term F<sub>n</sub> is of order 1 and is given in terms of the Jacobi theta function. Our theorem allows us to derive various precise results on the disk counting function. For example, we prove that the asymptotic fluctuations of the number of points in one component are of order 1 and are given by an oscillatory discrete Gaussian. Furthermore, the variance of this random variable enjoys asymptotics described by the Weierstrass ℘-function.</p>}},
  author       = {{Ameur, Yacin and Charlier, Christophe and Cronvall, Joakim and Lenells, Jonatan}},
  issn         = {{0001-8708}},
  keywords     = {{Moment generating functions; Oscillatory asymptotics; Random matrix theory}},
  language     = {{eng}},
  publisher    = {{Elsevier}},
  series       = {{Advances in Mathematics}},
  title        = {{Disk counting statistics near hard edges of random normal matrices : The multi-component regime}},
  url          = {{http://dx.doi.org/10.1016/j.aim.2024.109549}},
  doi          = {{10.1016/j.aim.2024.109549}},
  volume       = {{441}},
  year         = {{2024}},
}