Stability of Higher-Order Spacing Ratios for Poisson and Semi-Poisson Ensembles with Missing Levels

Abstract

We investigate the impact of random level loss on the higher-order level spacing ratio statistics, P^(k)(r), of Poisson and semi-Poisson ensembles. By employing the Wasserstein distance to quantify distributional changes, we show that the two ensembles exhibit different responses to missing levels. The statistics of the Poisson ensemble are stable, with the distributions remaining invariant in the presence of missing levels. The Wasserstein distance between complete and incomplete spectra remains at the level of the numerical noise floor. However, the semi-Poisson ensemble demonstrates substantial statistical instability. Eliminating levels alters the distributions, yielding a measurable discrepancy, assessed by Wasserstein distance, which is 100 times greater than the associated numerical noise. The folded ratio distribution, P^(k)(ř), is the most effective method for probing the instability signal, which, as we show, is significantly amplified at higher orders (k > 1). These findings clarify a key uncertainty in spectral analysis, wherein a degraded semi-Poisson spectrum may be erroneously identified as a Poisson one. The stability of a spectrum's statistics in response to the artificial removal of levels serves as a robust, model-independent test for differentiating between uncorrelated and correlated quantum systems, turning the problem of missing levels into a powerful diagnostic feature.