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This article is cited in 1 scientific paper (total in 1 paper)
Gaussian approximation of the distribution of strongly repelling particles on the unit circle
A. Soshnikova, Yu. Xub a University of California at Davis, Davis, CA, USA
b KTH, Stockholm, Sweden
Abstract:
In this paper, we consider a strongly repelling model of $n$ ordered particles
$\{e^{i \theta_j}\}_{j=0}^{n-1}$ with the density
$p({\theta_0},\dots, \theta_{n-1})=\frac{1}{Z_n} \exp
\big\{-\frac{\beta}{2}\sum_{j \ne k} \sin^{-2} \big(
\frac{\theta_j-\theta_k}{2}\big)\big\}$, $\beta>0$.
Let $\theta_j=2 \pi j/n+x_j/n^2+\mathrm{const}$ such that
$\sum_{j=0}^{n-1}x_j=0$. Define $\zeta_n (2 \pi j/n) =x_j/\sqrt{n}$, and extend
$\zeta_n$ piecewise linearly to $[0, 2 \pi]$. We prove the functional
convergence of $\zeta_n(t)$ to
$\zeta(t)=\sqrt{\frac{2}{\beta}} \operatorname{Re} \big( \sum_{k=1}^{\infty}
\frac{1}{k} e^{ikt} Z_k \big)$,
where $Z_k$ are independent identically distributed complex standard Gaussian
random variables.
Keywords:
strongly repelling particles, multivariate Gaussian distribution, convergence of finite dimensional distributions, functional convergence.
Received: 25.03.2019 Revised: 08.11.2019 Accepted: 21.11.2019
Citation:
A. Soshnikov, Yu. Xu, “Gaussian approximation of the distribution of strongly repelling particles on the unit circle”, Teor. Veroyatnost. i Primenen., 65:4 (2020), 746–777; Theory Probab. Appl., 65:4 (2021), 588–615
Linking options:
https://www.mathnet.ru/eng/tvp5302https://doi.org/10.4213/tvp5302 https://www.mathnet.ru/eng/tvp/v65/i4/p746
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