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Gaussian Fluctuations and Moderate Deviations of Eigenvalues in Unitary Invariant Ensembles

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  • Deng Zhang

    (Shanghai Jiao Tong University)

Abstract

We study the limiting behavior of the k-th eigenvalue $$x_k$$ x k of unitary invariant ensembles with Freud-type and uniform convex potentials. As both k and $$n-k$$ n - k tend to infinity, we obtain Gaussian fluctuations for $$x_k$$ x k in the bulk and soft edge cases, respectively. Multi-dimensional central limit theorems, as well as moderate deviations, are also proved. This work generalizes earlier results in the GUE and unitary invariant ensembles with monomial potentials of even degree. In particular, we obtain the precise asymptotics of corresponding Christoffel–Darboux kernels as well.

Suggested Citation

  • Deng Zhang, 2019. "Gaussian Fluctuations and Moderate Deviations of Eigenvalues in Unitary Invariant Ensembles," Journal of Theoretical Probability, Springer, vol. 32(4), pages 1647-1687, December.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:4:d:10.1007_s10959-019-00939-4
    DOI: 10.1007/s10959-019-00939-4
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    References listed on IDEAS

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    1. Deng Zhang, 2017. "Tridiagonal Random Matrix: Gaussian Fluctuations and Deviations," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1076-1103, September.
    2. Hanna Döring & Peter Eichelsbacher, 2013. "Moderate Deviations via Cumulants," Journal of Theoretical Probability, Springer, vol. 26(2), pages 360-385, June.
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