Power spectra and autocovariances of level spacings beyond the Dyson conjecture

Introduced in the early days of random matrix theory, the autocovariances $\delta I_k^j=\mathrm{cov}(s_j,s_{j+k})$ of level spacings $\left\{s_j\right\}$ 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 $\delta I_k^j\approx -1/\beta {\pi }^2{k}^2$, 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 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.

Figure 1

Visualization of numerical data produced from $M={10}^8$ samples of $\mathrm{CUE}\left(N\right)$ with $N=1024$. Blue circles and red crosses represent the $k$ dependence of the differences $\delta {\mathcal{I}}_k(N,M)-\delta I_k^D$ (blue) and $\delta {\mathcal{I}}_k(N,M)-(\delta I_k^D+\delta I_k^S)$ (red) between numerically computed autocovariances $\delta {\mathcal{I}}_k(N,M)$ and either the leading-order Dyson's term [Eq. (7)] or the improved analytical prediction [Eq. (11)]. The vertical bars around the data points are $99\%$ confidence intervals. The inset shows the ratios $\delta {\mathcal{I}}_k(N,M)/\delta I_k^D$ and $\delta {\mathcal{I}}_k(N,M)/(\delta I_k^D+\delta I_k^S)$, respectively, for the same numerical data. A detailed description of the numerical procedure is given at the end of the Letter.