Asymptotic self-averaging at half filling

Determine whether the slow half-filling decay observed for the relative realization-to-realization variance \chi_q of the single-realization fluctuation ratio q_U persists asymptotically, and thereby establish its limiting exponent.

Background

The ensemble value q_ens is known exactly, but the paper also studies q_U for a single Haar eigenvector realization. The relative variance \chi_q decreases with system size, with substantially slower apparent decay at half filling than at fixed determinant depth.

Numerical data over accessible sizes suggest a half-filling behavior near N{-0.5}, but the authors emphasize that these sizes do not determine the asymptotic law. The unresolved problem is to establish the eventual self-averaging exponent or limiting decay behavior at half filling.

References

At half filling the measured variance grows from $0.8$ at $N=6$ to $6.2$ at $N=16$, approximately as $N2$ over this range, so that the absolute fluctuation strength increases while the ensemble value is approached in relative terms. The apparent exponent should not be read as asymptotic: over these sizes the exact half-filling value of $q_{\rm ens}$ has not yet reached its large-$N$ growth $q_{\rm ens}\sim N2/32$, and extrapolating that growth together with the fitted $\chi_q\propto N{-1/2}$ would give $Var_U(q_U)\propto N{7/2}$, so the accessible sizes do not fix the asymptotic exponent.

— Eigenstate thermalization beyond the envelope: exact two-point overlap statistics in random free fermions  (2609.17037 - Huang, 15 Sep 2026) in Section 5, “Numerical checks,” subsection discussing Eq. (chiq)