Scaling of the Reduced Energy Spectrum of Random Matrix Ensemble

We study the reduced energy spectrum $\{E_i^{(n)}\}$, which is constructed by picking one level from every $n$ levels of the original spectrum $\{E_i\}$, in a Gaussian ensemble of random matrix with Dyson index $\beta\in \left( 0,\infty \right) $\mathrm{.}I{t}^{\mathrm{\prime }}sshown${E_{i}^{(n)}}$ bears the same form of probability distribution as $\{E_i\}$ with a rescaled parameter $\gamma =\frac{n(n+1)}{2}\beta +n-1$\mathrm{.}Notably,the$n$-th order level spacing and non-overlapping gap ratio in $\{E_i\}$ become the lowest-order ones in $\{E_i^{(n)}\}$, hence their distributions will rescale in an identical way. Numerical evidences are provided by simulating random spin chain as well as modelling random matrices. Our results establish the higher-order spacing distributions in random matrix ensembles beyond GOE,GUE,GSE, and reveals a hierarchy of structures hidden in the energy spectrum.