Circular law and arc law for truncation of random unitary matrix

Zhishan Dong, Tiefeng Jiang, Danning Li

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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 languageEnglish (US)
Article number013301
JournalJournal of Mathematical Physics
Volume53
Issue number1
DOIs
StatePublished - Jan 4 2012

Bibliographical note

Funding Information:
We thank Professor Sho Matsumoto for a very useful comment on the proof of Theorem 1. The research of Zhishan Dong was supported in part by National Science Foundation of China (NSFC) Grant No. 11001104 and the research of Tiefeng Jiang was supported in part by NSF FRG Grant DMS-0449365.

Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

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