A point process on the unit circle with antipodal interactions

Abstract

We introduce the point process \begin{align*} \frac{1}{Z_{n}}\prod_{1 \leq j < k \leq n} |e^{iθ_{j}}+e^{iθ_{k}}|^β\prod_{j=1}^{n} dθ_{j}, \qquad θ_{1},\ldots,θ_{n} \in (-π,π], \quad β> 0, \end{align*} 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${\hat{I}}^2$E involves $n$ uniformly distributed random variables on the unit circle that repel each other.) We consider linear statistics of the form ${\sum }_{j=1}^{n}g(\hat{I}{¸}_j)$ as $n\to \infty$, where $g\in C^{1,q}$ and $2\ddot{I}€$-periodic. We prove that the leading order fluctuations around the mean are of order $n$ and given by $(g(U)-{\int }_{-\ddot{I}€}^{\ddot{I}}€g(\hat{I}¸)\frac{d\hat{I}¸}{2\ddot{I}€})n$, where $U\sim \mathrm{Uniform}(-\ddot{I}€,\ddot{I}€]$. We also prove that the subleading fluctuations around the mean are of order $\sqrt{n}$ and of the form ${\mathcal{N}}_{\mathbb{R}}(0,4g^{\prime }(U{)}^2/{\hat{I}}^2)\sqrt{n}$, 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 [8,6] to obtain asymptotics of related $n$-fold integrals.

Results are improved; 12 pages, 1 figure