Superposition and higher-order spacing ratios in random matrix theory with application to complex systems

The statistical properties of spectra quantum systems within the framework random matrix theory is widely used in many areas physics. These are affected, if two or more sets superposed, resulting from discrete symmetries present system. Superposition $m$ such circular orthogonal, unitary and symplectic ensembles studied numerically using higher-order spacing ratios. For given Dyson index $\beta$, modified $\beta'$ tabulated whose nearest neighbor distribution identical to that $k$-th order ratio. case $m=2$ ($m=3$) COE (CUE) a scaling relation between $k$ given. COE, it conjectured for $k=m+1$ ($m\geq2$) $k=m-3$-th ($m\geq5$) ratio $m+2$ $m-4$ respectively. Whereas CSE, $k=m-1$-th ($m\geq3$) $2m+3$ $2(m-2)$ We also conjecture ($k$) sequence as function ($m$) unique. Strong numerical evidence support these results presented. tested on complex like measured nuclear resonances, chaotic kicked top spin chains.