Dyson猜想之外能级间距的功率谱和自协方差
Power spectra and autocovariances of level spacings beyond the Dyson conjecture
摘要：
在随机矩阵理论的早期引入的，水平间距的自协方差$δI^j_k={\rm cov}(s_j, s_{j+k})$ $\{s_j\}$提供了关于个体本征水平之间的相关性的详细信息。Dyson首先指出无限维随机矩阵展开谱中远本征能级的自协方差应呈现幂律衰减$δI^j_k\approx -1/βπ^2k^2$，其中$β$是对称指数。在这封信中，我们建立了一个确切的联系之间的自协方差的水平间距和他们的功率谱，并表明，对于$β=2$，后者承认表示在第五Painlevé超越。这个结果被进一步利用，以确定一个渐近展开的自协方差，再现戴森公式，以及提供subleading的更正it. High-precision数值模拟借给独立支持我们的结果。
英文摘要：
Introduced in the early days of random matrix theory, the autocovariances $δI^j_k={\rm cov}(s_j, s_{j+k})$ 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 $δI^j_k\approx -1/βπ^2k^2$, 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.
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