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A point process on the unit circle with antipodal interactions

Charlier, Christophe LU (2025) In Journal of Approximation Theory 310.
Abstract

We introduce the point process [Formula presented] where Zn is the normalization constant. This point process is attractive: it involves n dependent, uniformly distributed random variables on the unit circle that attract each other. (For comparison, the well-studied CβE involves n uniformly distributed random variables on the unit circle that repel each other.) We consider linear statistics of the form ∑j=1ng(θj) as n→∞, where g∈C1,q and 2π-periodic. We prove that the leading order fluctuations around the mean are of order n and given by [Formula presented], where U∼Uniform(−π,π]. We also prove that the subleading fluctuations around the mean are of order n and of the form... (More)

We introduce the point process [Formula presented] where Zn is the normalization constant. This point process is attractive: it involves n dependent, uniformly distributed random variables on the unit circle that attract each other. (For comparison, the well-studied CβE involves n uniformly distributed random variables on the unit circle that repel each other.) We consider linear statistics of the form ∑j=1ng(θj) as n→∞, where g∈C1,q and 2π-periodic. We prove that the leading order fluctuations around the mean are of order n and given by [Formula presented], where U∼Uniform(−π,π]. We also prove that the subleading fluctuations around the mean are of order n and of the form NR(0,4g′(U)2/β)n, i.e. that the subleading fluctuations are given by a Gaussian random variable that itself has a random variance. Our proof uses techniques developed by McKay and Isaev (McKay, 1990; Isaev and McKay, 2018) to obtain asymptotics of related n-fold integrals.

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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Asymptotics, Attractive interactions, Point processes, Smooth statistics
in
Journal of Approximation Theory
volume
310
article number
106161
publisher
Academic Press
external identifiers
  • scopus:86000322250
ISSN
0021-9045
DOI
10.1016/j.jat.2025.106161
language
English
LU publication?
yes
id
895060b8-69e7-4ca0-8375-2d37f84b7838
date added to LUP
2025-06-10 09:49:00
date last changed
2025-06-10 09:49:35
@article{895060b8-69e7-4ca0-8375-2d37f84b7838,
  abstract     = {{<p>We introduce the point process [Formula presented] where Z<sub>n</sub> is the normalization constant. This point process is attractive: it involves n dependent, uniformly distributed random variables on the unit circle that attract each other. (For comparison, the well-studied CβE involves n uniformly distributed random variables on the unit circle that repel each other.) We consider linear statistics of the form ∑<sub>j=1</sub><sup>n</sup>g(θ<sub>j</sub>) as n→∞, where g∈C<sup>1,q</sup> and 2π-periodic. We prove that the leading order fluctuations around the mean are of order n and given by [Formula presented], where U∼Uniform(−π,π]. We also prove that the subleading fluctuations around the mean are of order n and of the form N<sub>R</sub>(0,4g<sup>′</sup>(U)<sup>2</sup>/β)n, i.e. that the subleading fluctuations are given by a Gaussian random variable that itself has a random variance. Our proof uses techniques developed by McKay and Isaev (McKay, 1990; Isaev and McKay, 2018) to obtain asymptotics of related n-fold integrals.</p>}},
  author       = {{Charlier, Christophe}},
  issn         = {{0021-9045}},
  keywords     = {{Asymptotics; Attractive interactions; Point processes; Smooth statistics}},
  language     = {{eng}},
  publisher    = {{Academic Press}},
  series       = {{Journal of Approximation Theory}},
  title        = {{A point process on the unit circle with antipodal interactions}},
  url          = {{http://dx.doi.org/10.1016/j.jat.2025.106161}},
  doi          = {{10.1016/j.jat.2025.106161}},
  volume       = {{310}},
  year         = {{2025}},
}