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) |
|---|---|
| Article number | 013301 |
| Journal | Journal of Mathematical Physics |
| Volume | 53 |
| Issue number | 1 |
| DOIs | |
| State | Published - 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.