Scaling of the Reduced Energy Spectrum of Random Matrix Ensemble

TL;DR

This paper investigates how the reduced energy spectrum derived from Gaussian random matrix ensembles scales, revealing a hierarchy of spectral structures and providing numerical evidence through simulations of spin chains and random matrices.

Contribution

It introduces a scaling relation for higher-order energy level spacings in Gaussian ensembles, extending understanding beyond classical GOE, GUE, and GSE cases.

Findings

Higher-order spacing distributions rescale with a specific parameter.

Reduced spectra preserve the form of the original distribution.

Numerical simulations support the theoretical scaling relations.

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.