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$β$E involves $n$ uniformly distributed random variables on the unit circle that repel each other.) We consider linear statistics of the form $\sum_{j=1}^{n}g(θ_{j})$ as $n \to \infty$, where $g\in C^{1,q}$ and $2π$-periodic. We prove that the leading order fluctuations around the mean are of order $n$ and given by $\smash{\big(g(U)-\int_{-π}^πg(θ) \frac{dθ}{2π}}\big)n$, where $U \sim \mathrm{Uniform}(-π,π]$. We also prove that the subleading fluctuations around the mean are of order $\sqrt{n}$ and of the form $\mathcal{N}_{\mathbb{R}}(0,4g'(U)^{2}/β)\sqrt{n}$, 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.