اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدGaussian Approximation of the Distribution of Strongly Repelling Particles on the Unit Circle
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In this paper, we consider a strongly-repelling model of $n$ ordered particles ${e^{i theta_j}}_{j=0}^{n-1}$ with the density $p({theta_0},cdots, theta_{n-1})=frac{1}{Z_n} exp left{-frac{beta}{2}sum_{j eq k} sin^{-2} left( frac{theta_j-theta_k}{2}right)right}$, $beta>0$. 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.
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