Variance asymptotics and Gaussian fluctuations for a singular pair statistic of the sine process

This article studies an unordered pair statistic, \(V_L\), of the sine-kernel determinantal point process. On the deterministic interval \(I_L=[-L/2,L/2]\), the pair interaction potential is defined by the function \(u\mapsto u^{-2}\), to which we apply a distance cutoff \(\varepsilon>0\). For every \(\varepsilon>0\), the observable under consideration is \[V_L(\varepsilon)=\sum_{\substack{x,y\in X\cap I_L\\ x \varepsilon\}}}{(x-y)^2}.\] We first consider the untruncated regime, corresponding to \(\varepsilon=0\). We prove that \(V_L(0)<\infty\) almost surely, that \(V_L(0)\in L^1(\mathbb P)\setminus L^2(\mathbb P)\), and that \[\lim_{L\to\infty}\mathbb E V_L(0)/L=\pi^2/3.\] For every \(\varepsilon>0\), we then establish the existence and strict positivity of the asymptotic variance: \[\lim_{L\to\infty}L^{-1}\operatorname{Var}(V_L(\varepsilon))=\sigma_\varepsilon^2\in(0,\infty).\] Moreover, a spectral reduction of the variance yields the asymptotic expansion as \(\varepsilon\downarrow0\), \[\sigma_\varepsilon^2=\frac{\pi^2}{3\varepsilon}-\frac{\pi^4}{5}+\frac{22\pi^4}{45}\varepsilon+o(\varepsilon).\] Finally, for every \(\varepsilon>0\), we establish a central limit theorem for the centered observable normalized by \(\sqrt L\). Taking the successive limits \(L\to\infty\) and then \(\varepsilon\downarrow0\) in the iterated regime yields the Gaussian limit: \[\begin{aligned}&\sqrt{\frac{\varepsilon}{L}}(V_L(\varepsilon)-\mathbb E V_L(\varepsilon))\xrightarrow[L\to\infty]{d}\mathcal N(0,\varepsilon\sigma_\varepsilon^2),\\[4pt]&\mathcal N(0,\varepsilon\sigma_\varepsilon^2)\xrightarrow[\varepsilon\downarrow0]{w}\mathcal N\left(0,\frac{\pi^2}{3}\right).\end{aligned}\]