Local box-counting dimensions of discrete quantum eigenvalue spectra: Analytical connection to quantum spectral statistics

TL;DR

This paper derives an explicit formula linking the local box-counting dimension of discrete quantum spectra to their nearest-neighbor spacing distribution, validating a two-decade-old hypothesis and applying it to various spectral models.

Contribution

It provides the first explicit analytical connection between local box-counting dimensions and NNSD, including formulas for Poisson, GOE, GUE, and GSE spectra, with validation against numerical data.

Findings

Excellent agreement between theory and numerical data for Poisson and GOE spectra.

Derived formulas accurately describe local box-counting dimensions of Riemann zeta zeros.

Validated the hypothesis that local box-counting dimension depends solely on NNSD.

Abstract

Two decades ago, Wang and Ong [Phys. Rev. A 55, 1522 (1997)] hypothesized that the local box-counting dimension of a discrete quantum spectrum should depend exclusively on the nearest-neighbor spacing distribution (NNSD) of the spectrum. In this paper, we validate their hypothesis by deriving an explicit formula for the local box-counting dimension of a countably-infinite discrete quantum spectrum. This formula expresses the local box-counting dimension of a spectrum in terms of single and double integrals of the NNSD of the spectrum. As applications, we derive an analytical formula for Poisson spectra and closed-form approximations to the local box-counting dimension for spectra having Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE), and Gaussian symplectic ensemble (GSE) spacing statistics. In the Poisson and GOE cases, we compare our theoretical formulas with the published numerical data of Wang and Ong and observe excellent agreement between their data and our theory. We also study numerically the local box-counting dimensions of the Riemann zeta function zeros and the alternate levels of GOE spectra, which are often used as numerical models of spectra possessing GUE and GSE spacing statistics, respectively. In each case, the corresponding theoretical formula is found to accurately describe the numerically-computed local box-counting dimension.