Edge scaling limits for a family of non-Hermitian random matrix ensembles

Details Edge scaling limits for a family of non-Hermitian random matrix ensembles Edge scaling limits for a family of non-Hermitian random matrix ensembles

2008-08-19

arxiv.org/abs/0808.2608v1

Probab. Theory Relat Fields, 147, 241--271 (2010)

Mathematics

Probability

44 pages

Scientific paper

10.1007/s00440-009-0207-9

A family of random matrix ensembles interpolating between the GUE and the Ginibre ensemble of $n\times n$ matrices with iid centered complex Gaussian entries is considered. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order $n^{-1/3}$. In this regime, the family of limiting probability distributions of the maximum of the real parts of the eigenvalues interpolates between the Gumbel and Tracy-Widom distributions.

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