Power spectra and autocovariances of level spacings beyond the Dyson conjecture

TL;DR

This paper establishes a precise mathematical relationship between level spacing autocovariances and their power spectrum in random matrix theory, revealing new asymptotic behaviors and confirming Dyson's conjecture through analytical and numerical methods.

Contribution

It provides an exact link between autocovariances and power spectrum, expressing the latter via a Painlevé transcendent, and derives subleading corrections to Dyson's power-law decay.

Findings

Derived an asymptotic expansion for autocovariances

Expressed power spectrum in terms of Painlevé transcendent

Supported results with high-precision numerical simulations

Abstract

Introduced in the early days of random matrix theory, the autocovariances $\delta I^j_k={\rm cov}(s_j, s_{j+k})$ of level spacings $\{s_j\}$ accommodate a detailed information on correlations between individual eigenlevels. It was first conjectured by Dyson that the autocovariances of distant eigenlevels in the unfolded spectra of infinite-dimensional random matrices should exhibit a power-law decay $\delta I^j_k\approx -1/\beta\pi^2k^2$, where $\beta$ is the symmetry index. In this Letter, we establish an exact link between the autocovariances of level spacings and their power spectrum, and show that, for $\beta=2$, the latter admits a representation in terms of a fifth Painlev\'e transcendent. This result is further exploited to determine an asymptotic expansion for autocovariances that reproduces the Dyson formula as well as provides the subleading corrections to it. High-precision numerical simulations lend independent support to our results.