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.

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• Complex systems
• Mathematical physics
• Quantum chaos
• 0-dimensional systems
• Disordered systems
• Quantum billiards
• Data analysis
• Integrable systems
• Random matrix theory