Circular law and arc law for truncation of random unitary matrix

Zhishan Dong, Tiefeng Jiang, Danning Li

Research output: Contribution to journalArticle

16 Scopus citations

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

Fingerprint Dive into the research topics of 'Circular law and arc law for truncation of random unitary matrix'. Together they form a unique fingerprint.

  • Cite this