Gaussian Fluctuations and Moderate Deviations of Eigenvalues in Unitary Invariant Ensembles

Deng Zhang ()
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Deng Zhang: Shanghai Jiao Tong University

Journal of Theoretical Probability, 2019, vol. 32, issue 4, 1647-1687

Abstract: 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.

Keywords: Gaussian fluctuations; Moderate deviation principle; Riemann–Hilbert approach; Unitary invariant ensembles; 60B20; 60F05; 60F10 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10959-019-00939-4

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