Higher-order level spacings in random matrix theory based on Wigner's conjecture

TL;DR

This paper analytically derives the distribution of higher order level spacings in random matrix theory, revealing generalized distributions for Gaussian and Poisson ensembles, supported by numerical simulations including quantum spin systems and Riemann zeros.

Contribution

It introduces a generalized framework for higher order level spacings in random matrices, extending Wigner's conjecture to these distributions with analytical formulas and numerical validation.

Findings

Higher order spacings in Gaussian ensembles follow a generalized Wigner-Dyson distribution.

Poisson ensemble spacings follow a generalized semi-Poisson distribution.

Numerical simulations confirm the analytical results using spin systems and Riemann zeta zeros.

Abstract

The distribution of higher order level spacings, i.e. the distribution of $\{s_{i}^{(n)}=E_{i+n}-E_{i}\}$ with $n\geq 1$ is derived analytically using a Wigner-like surmise for Gaussian ensembles of random matrix as well as Poisson ensemble. It is found $s^{(n)}$ in Gaussian ensembles follows a generalized Wigner-Dyson distribution with rescaled parameter $\alpha=\nu C_{n+1}^2+n-1$, while that in Poisson ensemble follows a generalized semi-Poisson distribution with index $n$. Numerical evidences are provided through simulations of random spin systems as well as non-trivial zeros of Riemann zeta function. The higher order generalizations of gap ratios are also discussed.