### 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 |

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### Keywords

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

### Cite this

**Spectral Radii of Large Non-Hermitian Random Matrices.** / Jiang, Tiefeng; Qi, Yongcheng.

Research output: Contribution to journal › Article

*Journal of Theoretical Probability*, vol. 30, no. 1, pp. 326-364. https://doi.org/10.1007/s10959-015-0634-8

}

TY - JOUR

T1 - Spectral Radii of Large Non-Hermitian Random Matrices

AU - Jiang, Tiefeng

AU - Qi, Yongcheng

PY - 2017/3/1

Y1 - 2017/3/1

N2 - 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.

AB - 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.

KW - Determinantal point process

KW - Eigenvalue

KW - Extreme value

KW - Independence

KW - Non-Hermitian random matrix

KW - Spectral radius

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U2 - 10.1007/s10959-015-0634-8

DO - 10.1007/s10959-015-0634-8

M3 - Article

AN - SCOPUS:84939600653

VL - 30

SP - 326

EP - 364

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

IS - 1

ER -