## Abstract

By using the independence structure of points following a determinantal point process, we study the radii of the spherical ensemble, the truncation of the circular unitary ensemble and the product ensemble with parameters n and k. The limiting distributions of the three radii are obtained. They are not the Tracy–Widom distribution. In particular, for the product ensemble, we show that the limiting distribution has a transition phenomenon: When k/ n→ 0 , k/ n→ α∈ (0 , ∞) and k/ n→ ∞, the liming distribution is the Gumbel distribution, a new distribution μ and the logarithmic normal distribution, respectively. The cumulative distribution function (cdf) of μ is the infinite product of some normal distribution functions. Another new distribution ν is also obtained for the spherical ensemble such that the cdf of ν is the infinite product of the cdfs of some Poisson-distributed random variables.

Original language | English (US) |
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Pages (from-to) | 326-364 |

Number of pages | 39 |

Journal | Journal of Theoretical Probability |

Volume | 30 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2017 |

### Bibliographical note

Funding Information:The research of Tiefeng Jiang was supported in part by NSF Grant DMS-1209166 and DMS-1406279.

Funding Information:

The research of Yongcheng Qi was supported in part by NSF Grant DMS-1005345.

Publisher Copyright:

© 2015, Springer Science+Business Media New York.

## Keywords

- Determinantal point process
- Eigenvalue
- Extreme value
- Independence
- Non-Hermitian random matrix
- Spectral radius