Universal scaling of higher-order spacing ratios in Gaussian random matrices

TL;DR

This paper analytically derives a universal scaling relation for higher-order spacing ratios in Gaussian random matrices, confirming earlier numerical observations and extending understanding of spectral statistics in large random matrices.

Contribution

The paper provides the first analytical proof of the universal scaling relation for higher-order spacing ratios in Gaussian ensembles, validating previous numerical results.

Findings

Universal scaling relation proven analytically for $r^{(k)}\rightarrow0$ and $r^{(k)}\rightarrow\infty$

Extension of spectral statistics understanding in Gaussian random matrices

Confirmation of earlier numerical scaling observations

Abstract

Higher-order spacing ratios are investigated analytically using a Wigner-like surmise for Gaussian ensembles of random matrices. For $k$-th order spacing ratio $(r^{(k)}$, $k>1)$ the matrix of dimension $2k+1$ is considered. A universal scaling relation for this ratio, known from earlier numerical studies, is proved in the asymptotic limits of $r^{(k)}\rightarrow0$ and $r^{(k)}\rightarrow \infty$.