A point process on the unit circle with antipodal interactions

TL;DR

This paper introduces a new point process on the unit circle with attractive interactions, analyzes the fluctuations of linear statistics, and finds that the leading order is deterministic while the subleading order is Gaussian with random variance.

Contribution

It presents a novel attractive point process on the circle and characterizes the asymptotic fluctuations of linear statistics, including a Gaussian subleading term with random variance.

Findings

Leading order fluctuations are deterministic and proportional to n.

Subleading fluctuations are Gaussian with variance depending on the random variable U.

The process contrasts with the well-known repulsive CβE by exhibiting attraction.

Abstract

We introduce the point process \begin{align*} \frac{1}{Z_{n}}\prod_{1 \leq j < k \leq n} |e^{i\theta_{j}}+e^{i\theta_{k}}|^{\beta}\prod_{j=1}^{n} d\theta_{j}, \qquad \theta_{1},\ldots,\theta_{n} \in (-\pi,\pi], \quad \beta > 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$\beta$E involves $n$ uniformly distributed random variables on the unit circle that repel each other.) We consider linear statistics of the form $\sum_{j=1}^{n}g(\theta_{j})$ 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 $\smash{\big(g(U)-\int_{-\pi}^{\pi}g(\theta) \frac{d\theta}{2\pi}}\big)n$, 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 $\mathcal{N}_{\mathbb{R}}(0,4g'(U)^{2}/\beta)\sqrt{n}$, 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 [8,6] to obtain asymptotics of related $n$-fold integrals.