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 $p({\theta_0},\cdots, \theta_{n-1})=\frac{1}{Z_n} \exp \left\{-\frac{\beta}{2}\sum_{j \neq k} \sin^{-2} \left( \frac{\theta_j-\theta_k}{2}\right)\right\}$, $\beta>0$. Let $\theta_j=\frac{2 \pi j}{n}+\frac{x_j}{n^2}+const$ such that $\sum_{j=0}^{n-1}x_j=0$. Define $\zeta_n \left( \frac{2 \pi j}{n}\right) =\frac{x_j}{\sqrt{n}}$ and extend $\zeta_n$ piecewise linearly to $[0, 2 \pi]$. We prove the functional convergence of $\zeta_n(t)$ to $\zeta(t)=\sqrt{\frac{2}{\beta}} \mathfrak{Re} \left( \sum_{k=1}^{\infty} \frac{1}{k} e^{ikt} Z_k \right)$, where $Z_k$ are i.i.d. complex standard Gaussian random variables.