### Abstract

Let V be the m × m upper-left corner of an n × n Haar-invariant unitary matrix. Let λ _{1},. ., λ _{m} be the eigenvalues of V. We prove that the empirical distribution of a normalization of λ _{1},. ., λ _{m} goes to the circular law, that is, the uniform distribution on {z ε C; |z| = 1} as m → 8 with m/n → 0. We also prove that the empirical distribution of λ _{1},. ., λ _{m} goes to the arc law, that is, the uniform distribution on {z ε C; |z| = 1} as m/n → 1. These explain two observations by ? Zyczkowski and Sommers (2000).

Original language | English (US) |
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Article number | 013301 |

Journal | Journal of Mathematical Physics |

Volume | 53 |

Issue number | 1 |

DOIs | |

State | Published - Jan 4 2012 |

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## Cite this

Dong, Z., Jiang, T., & Li, D. (2012). Circular law and arc law for truncation of random unitary matrix.

*Journal of Mathematical Physics*,*53*(1), [013301]. https://doi.org/10.1063/1.3672885