A point process on the unit circle with antipodal interactions

TL;DR

This work studies an attractive, antipodal-interaction point process on the unit circle, defined by a density proportional to $\prod_{j<k}|e^{i\theta_j}+e^{i\theta_k}|^{\beta}$, in contrast to the repulsive circular $\beta$-ensemble. For smooth $2\pi$-periodic test functions $g$, the authors analyze linear statistics $\sum_{j=1}^n g(\theta_j)$ as $n\to\infty$ and prove a two-scale fluctuation description: the leading term is linear in $n$, with random mean determined by $U\sim\mathrm{Uniform}(-\pi,\pi]$, i.e. $n\big(g(U)-\int_{-\pi}^{\pi} g(\theta)\frac{d\theta}{2\pi}\big)$, and the subleading term is of order $\sqrt{n}$ and Gaussian with random variance $4g'(U)^2/\beta$. The analysis relies on McKay–Isaev type asymptotics for related $n$-fold integrals to obtain precise generating-function expansions, and it highlights a markedly different fluctuation structure from the classical C$\beta$E, reflecting the attractive nature of the antipodal interactions. The results imply a concentration of the empirical measure on a shrinking arc and reveal a novel interaction between randomness from $U$ and the local derivatives of $g$ in the fluctuations. These insights contribute to the understanding of attractive point processes on the circle and suggest potential transitions when $\beta$ scales with $n$. Overall, the paper combines rigorous asymptotics with probabilistic interpretations to characterize smooth linear statistics in this attractive setting.

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.

Figures (1)

• Figure 1: Illustration of the point process \ref{['new point process intro']} with $n=50$. With high probability all points are close to each other.

Theorems & Definitions (7)

• Theorem 1.1
• Theorem 1.2
• proof
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof