A point process on the unit circle with mirror-type interactions

TL;DR

This paper analyzes a mirror-interaction point process on the unit circle, revealing diverse asymptotic fluctuation behaviors of linear statistics and providing detailed large-sample asymptotics for the normalization constant.

Contribution

It introduces and studies a novel point process with mirror-type interactions, characterizing its complex fluctuation regimes and deriving precise asymptotics for its normalization constant.

Findings

Fluctuations can be Bernoulli, Gaussian, or mixed, depending on the test function g.

The process exhibits diverse asymptotic fluctuation orders: n, 1, or o(1).

Large n asymptotics for the normalization constant Z_n are obtained, including the order 1 term.

Abstract

We consider 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. The feature of this process is that the points $e^{i\theta_{1}},\ldots,e^{i\theta_{n}}$ interact with the mirror points reflected over the real line $e^{-i\theta_{1}},\ldots,e^{-i\theta_{n}}$. We study smooth linear statistics of the form $\sum_{j=1}^{n}g(\theta_{j})$ as $n \to \infty$, where $g$ is $2\pi$-periodic. We prove that a wide range of asymptotic scenarios can occur: depending on $g$, the leading order fluctuations around the mean can (i) be of order $n$ and purely Bernoulli, (ii) be of order $1$ and purely Gaussian, (iii) be of order $1$ and purely Bernoulli, or (iv) be of order $1$ and of the form $BN_{1}+(1-B)N_{2}$, where $N_{1},N_{2}$ are two independent Gaussians and $B$ is a Bernoulli that is independent of $N_{1}$ and $N_{2}$. The above list is not exhaustive: the fluctuations can be of order $n$, of order $1$ or $o(1)$, and other random variables can also emerge in the limit. We also obtain large $n$ asymptotics for $Z_{n}$ (and some generalizations), up to and including the term of order $1$. Our proof is inspired by a method developed by McKay and Wormald [12] to estimate related $n$-fold integrals.