Log-normal Distribution of Level Curvatures in the Localized Regime: Analytical Verification

We study numerically and analytically the moments of the dimensionless level curvature for one-dimensional disordered rings of the circumference L pierced by a magnetic flux. The negative moments of the curvature distribution can be evaluated analytically in the extreme localization limit. The ensuing small curvature asymptotic of the corresponding distribution has a "log-normal" behavior. Numerically studied positive moments show differences from other log-normally distributed quantities.