Scaling of the Reduced Energy Spectrum of Random Matrix Ensemble

Abstract: 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) $. It's shown ${E_{i}^{(n)}}$ bears the same form of probability distribution as ${E_{i}}$ with a rescaled parameter $\gamma =\frac{n(n+1)}{2}\beta +n-1$. 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.