A point process on the unit circle with antipodal interactions

Abstract

We introduce the point process

$\frac{1}{Z_n}{\prod }_{1\le j<k\le n}{|e^{i{\theta }_j}+e^{i{\theta }_k}|}^{\beta }{\prod }_{j=1}^{n}d{\theta }_j,{\theta }_1,\dots ,{\theta }_n\in (-\pi ,\pi ],\beta >0,$

where $Z_n$ 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 C$\beta$E involves $n$ uniformly distributed random variables on the unit circle that repel each other.) We consider linear statistics of the form ${\sum }_{j=1}^{n}g\left({\theta }_j\right)$ as $n\to \infty$, where $g\in C^{1,q}$ and $2\pi$-periodic. We prove that the leading order fluctuations around the mean are of order $n$ and given by $\left(g\left(U\right)-{\int }_{-\pi }^{\pi }g\left(\theta \right)\frac{d\theta }{2\pi }\right)n$, where $U\sim \mathrm{Uniform}(-\pi ,\pi ]$. We also prove that the subleading fluctuations around the mean are of order $\sqrt{n}$ and of the form $N_R(0,4g^{\prime }{\left(U\right)}^2/\beta )\sqrt{n}$, 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.