Gaussian approximation of the distribution of strongly repelling particles on the unit circle

In this paper, we consider a strongly repelling model of $n$ ordered particles $\{e^{i \theta_j}\}_{j=0}^{n-1}$ with the density $p({\theta_0},\dots, \theta_{n-1})=\frac{1}{Z_n} \exp \big\{-\frac{\beta}{2}\sum_{j \ne k} \sin^{-2} \big( \frac{\theta_j-\theta_k}{2}\big)\big\}$, $\beta>0$. Let $\theta_j=2 \pi j/n+x_j/n^2+\mathrm{const}$ such that $\sum_{j=0}^{n-1}x_j=0$. Define $\zeta_n (2 \pi j/n) =x_j/\sqrt{n}$, and extend $\zeta_n$ piecewise linearly to $[0, 2 \pi]$. We prove the functional convergence of $\zeta_n(t)$ to $\zeta(t)=\sqrt{\frac{2}{\beta}} \operatorname{Re} \big( \sum_{k=1}^{\infty} \frac{1}{k} e^{ikt} Z_k \big)$, where $Z_k$ are independent identically distributed complex standard Gaussian random variables.