Mesoscopic Eigenvalue Density Correlations of Wigner Matrices

Tuesday, 28 August, 2018

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The celebrated Wigner-Gaudin-Mehta-Dyson (WGMD) (or sine kernel) statistics of random matrix theory describes the universal correlations of eigenvalues on the microscopic scale, i.e. correlations of eigenvalue densities at points whose separation is comparable to the typical eigenvalue spacing. We investigate to what extent these statistics remain valid on mesoscopic scales, where the densities are measured at points whose separation is much larger than the typical eigenvalue spacing. In the mesoscopic regime, density-density correlations are much weaker than in the microscopic regime.
More precisely, we compute the connected two-point spectral correlation function of a Wigner matrix at two mesoscopically separated points. We obtain a precise and explicit formula for the two-point function. Among other things, this formula implies that the WGMD statistics are valid to leading order on all mesoscopic scales, that the real symmetric terms contain subleading corrections matching precisely the WGMD statistics, while in the complex Hermitian case these subleading corrections are absent. We also uncover non-universal subleading correlations, which dominate over the universal ones beyond a certain intermediate mesoscopic scale.
Our formula reproduces, in the extreme macroscopic regime, the well-known non-universal correlations of linear statistics on the macroscopic scale. Thus, our results bridges, within one continuous unifying picture, two previously unrelated classes of results in random matrix theory: correlations of linear statistics and microscopic eigenvalue universality.
The main technical achievement of the proof is the development of an algorithm that allows to compute asymptotic expansions up to arbitrary accuracy of arbitrary correlation functions of mesoscopic linear statistics for Wigner matrices. Its proof is based on a hierarchy of Schwinger-Dyson equations for [...]


Yukun He
Antti Knowles