Power spectra and autocovariances of level spacings beyond the Dyson conjecture

Introduced in the early days of random matrix theory, the autocovariances delta Iki = cov(si, si+k) of level spacings {si} accommodate detailed information on the correlations between individual eigenlevels. It was first conjec-tured by Dyson that the autocovariances of distant eigenlevels in the unfolded spectra of infinite-dimensional random matrices should exhibit a power-law decay delta Iki eta -1/beta pi 2k2, 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 Painleve 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.

Automated summary

This paper introduces a method of autocovariance in the early days of random matrix theory to investigate the correlations between individual eigenlevels. By establishing a link between autocovariance and power spectrum, the researchers find that for symmetry index 2, the power spectrum can be represented by a fifth Painleve transcendent. High-precision numerical simulations provide independent support to this result.