### Abstract

Let U _{m} be an m×m Haar unitary matrix and U _{[} _{m,n} _{]} be its n×n truncation. In this paper the large deviation is proven for the empirical eigenvalue density of U _{[} _{m,n} _{]} as m/n→λ and n→∞. The rate function and the limit distribution are given explicitly. U _{[} _{m,n} _{]} is the random matrix model of quq, where u is a Haar unitary in a finite von Neumann algebra, q is a certain projection and they are free. The limit distribution coincides with the Brown measure of the operator quq.

Original language | English |
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Pages (from-to) | 175-189 |

Number of pages | 15 |

Journal | Probability Theory and Related Fields |

Volume | 133 |

Issue number | 2 |

DOIs | |

Publication status | Published - Oct 2005 |

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### Keywords

- Free probability
- Haar unitary
- Joint eigenvalue distribution
- Large deviation
- Random matrices
- Random matrix model
- Rate function
- Truncated Haar unitary

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Statistics and Probability

### Cite this

**Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices.** / Petz, D.; Réffy, Júlia.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 133, no. 2, pp. 175-189. https://doi.org/10.1007/s00440-004-0420-5

}

TY - JOUR

T1 - Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices

AU - Petz, D.

AU - Réffy, Júlia

PY - 2005/10

Y1 - 2005/10

N2 - Let U m be an m×m Haar unitary matrix and U [ m,n ] be its n×n truncation. In this paper the large deviation is proven for the empirical eigenvalue density of U [ m,n ] as m/n→λ and n→∞. The rate function and the limit distribution are given explicitly. U [ m,n ] is the random matrix model of quq, where u is a Haar unitary in a finite von Neumann algebra, q is a certain projection and they are free. The limit distribution coincides with the Brown measure of the operator quq.

AB - Let U m be an m×m Haar unitary matrix and U [ m,n ] be its n×n truncation. In this paper the large deviation is proven for the empirical eigenvalue density of U [ m,n ] as m/n→λ and n→∞. The rate function and the limit distribution are given explicitly. U [ m,n ] is the random matrix model of quq, where u is a Haar unitary in a finite von Neumann algebra, q is a certain projection and they are free. The limit distribution coincides with the Brown measure of the operator quq.

KW - Free probability

KW - Haar unitary

KW - Joint eigenvalue distribution

KW - Large deviation

KW - Random matrices

KW - Random matrix model

KW - Rate function

KW - Truncated Haar unitary

UR - http://www.scopus.com/inward/record.url?scp=24644460599&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=24644460599&partnerID=8YFLogxK

U2 - 10.1007/s00440-004-0420-5

DO - 10.1007/s00440-004-0420-5

M3 - Article

AN - SCOPUS:24644460599

VL - 133

SP - 175

EP - 189

JO - Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

JF - Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

SN - 0178-8051

IS - 2

ER -