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 \(u↦ u⁻²\), to which we apply a distance cutoff \(ε>0\). For every \(ε>0\), the observable under consideration is \[V_L(ε)=∑_{{x,y∈ X∩ I_L\\ x ε\}}}{(x-y)^2}.\] We first consider the untruncated regime, corresponding to \(ε=0\). We prove that \(V_L(0)<∞\) almost surely, that \(V_L(0)∈ L^1( P) L^2( P)\), and that \[limL→∞ E V_L(0)/L=π^2/3.\] For every \(ε>0\), we then establish the existence and strict positivity of the asymptotic variance: \[limL→∞L⁻¹Var(V_L(ε))=σ_ε^2∈(0,∞).\] Moreover, a spectral reduction of the variance yields the asymptotic expansion as \(ε↓0\), \[σ_ε^2=π^2/3ε-π^4/5+22π^4/45ε+o(ε).\] Finally, for every \(ε>0\), we establish a central limit theorem for the centered observable normalized by \(√ L\). Taking the successive limits \(L→∞\) and then \(ε↓0\) in the iterated regime yields the Gaussian limit: \[{aligned}&√ε/L(V_L(ε)- E V_L(ε))[L→∞]{d} N(0,εσ_ε^2),\\[4pt]& N(0,εσ_ε^2)[ε↓0]{w} N(0,π^2/3).{aligned}\]