Power spectra and autocovariances of level spacings beyond the Dyson conjecture.

Abstract

Introduced in the early days of random matrix theory, the autocovariances δI_{k}^{j}=cov(s_{j},s_{j+k}) of level spacings {s_{j}} accommodate detailed information on the 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 δI_{k}^{j}≈-1/βπ^{2}k^{2}, where β 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 β=2, the latter admits a representation in terms of a fifth Painlevé 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.