Higher-order spacings in the superposed spectra of random matrices with comparison to spacing ratios and application to complex systems

arXiv:2510.00503v2 Announce Type: replace Abstract: Higher-order spacing statistics in the $m$ superposed spectra of circular random matrices of the same class are studied numerically. We conjecture that for given $m$ (or order $k$) and $\beta$, the sequence of modified Dyson index $\beta'(k)$ (or $\beta'(m)$) obtained using the sum of absolute differences between the cumulative distribution functions method (denoted as $D(\beta')$) is unique. Also, for a given $k$, the distribution tends to the corresponding $k$-th order Poisson statistics in the limit $m\rightarrow \infty$. The quantum chaotic kicked top model for various Hilbert space dimensions is studied, and it is found to satisfy our conjecture. This involves the numerical verification of $m=2$ case of COE results. Our result can be used as a tool for the characterization of a system and to determine the symmetry structure of the system without desymmetrization of the spectra. Additionally, the comparative study of the higher-order spacing and ratio distributions in both $m=1$ and $m=2$ cases of COE as well as GOE is performed within and across these ensembles numerically using the $D(\beta')$ method. This study is carried out both by varying the dimension and keeping the number of realizations constant, and vice-versa. The same asymptotic higher-order statistics are observed across COE and GOE in terms of a given spectral fluctuation measure. But, within a given ensemble of COE or GOE, the results of higher-order spacing and ratio distributions agree with each other only up to some lower $k$, and beyond that, they start deviating from each other. Further, the spectral fluctuations of the intermediate map of various dimensions are studied. Various important observations and discussions from the analysis of our extensive numerical computations are presented.