On the asymptotic duality of spectral variances in random matrix theory and the "1/6" formula

TL;DR

This paper proves an asymptotic relation between spectral variances in random matrix theory for the β=2 class, introduces a new sum rule, and extends findings to other classes with numerical support.

Contribution

It establishes a previously unknown sum rule for level spacing auto-covariances and proves an asymptotic duality relation in the β=2 class, extending to β=1 and β=4.

Findings

Proves asymptotic exactness of the spectral variance relation for β=2.

Derives a new sum rule for level spacing auto-covariances.

Numerical analysis supports the theoretical results.

Abstract

A "mysterious" relation between the number variance and the variance of the $L$-th ordered eigenvalue, first suggested by French et al. [Ann. Phys. 113, 277 (1978)], is revisited and proven to be asymptotically exact for the $\beta=2$ Dyson symmetry class. Central to the proof is a previously unknown sum rule for the level spacing auto-covariances. Its derivation hinges on our previous work on the power spectrum description of eigenvalue fluctuations in random matrix theory. Analytical results for $\beta=2$ are complemented by conjectural extensions to the $\beta=1$ and $\beta=4$ symmetry classes. Our findings are corroborated by a comprehensive numerical analysis.