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

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.