### Abstract

Let V= (v_{ij})_{n×n} be a circular orthogonal ensemble. In this paper, for 1≤ m ≤ o (n/log n), we give a bound for the tail probability of max_{1≤ i,j ≤ m} v_{ij} - (1/n) y′_{i}y_{j} , where Y= (y_{1}, y_{n}) is a certain n×n matrix whose entries are independent and identically distributed random variables with the standard complex normal distribution ℂN (0,1). In particular, this implies that, for a sequence of such matrices {V_{n} = (v_{ij}^{(n)})_{n×n}, n ≥ 1}, as n→∞, n v_{ij} (n) converges in distribution to ℂN (0,1) for any i ≥ 1,j ≥ 1 with i ≤ j and n v_{ii}^{(n)} converges in distribution to 2 · ℂN (0,1) for any i ≥ 1.

Original language | English (US) |
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Article number | 063302 |

Journal | Journal of Mathematical Physics |

Volume | 50 |

Issue number | 6 |

DOIs | |

State | Published - 2009 |

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## Cite this

Jiang, T. (2009). The entries of circular orthogonal ensembles.

*Journal of Mathematical Physics*,*50*(6), [063302]. https://doi.org/10.1063/1.3152217