Gaussian Approximation of the Distribution of Strongly Repelling Particles on the Unit Circle

Abstract

In this paper, we consider a strongly-repelling model of $n$ ordered particles $\{e^{i θ_j}\}_{j=0}^{n-1}$ with the density $p({θ_0},\cdots, θ_{n-1})=\frac{1}{Z_n} \exp \left\{-\fracβ{2}\sum_{j \neq k} \sin^{-2} \left( \frac{θ_j-θ_k}{2}\right)\right\}$, $β>0$. Let $θ_j=\frac{2 πj}{n}+\frac{x_j}{n^2}+const$ such that $\sum_{j=0}^{n-1}x_j=0$. Define $ζ_n \left( \frac{2 πj}{n}\right) =\frac{x_j}{\sqrt{n}}$ and extend $ζ_n$ piecewise linearly to $[0, 2 π]$. We prove the functional convergence of $ζ_n(t)$ to $ζ(t)=\sqrt{\frac{2}β} \mathfrak{Re} \left( \sum_{k=1}^{\infty} \frac{1}{k} e^{ikt} Z_k \right)$, where $Z_k$ are i.i.d. complex standard Gaussian random variables.

35 pages. Some typos are corrected and the proof of Corollary 13 is polished