Circular Law and Arc Law for Truncation of Random Unitary Matrix
نویسندگان
چکیده
Let V be the m×m upper-left corner of an n× n Haar-invariant unitary matrix. Let λ1, · · · , λm be the eigenvalues of V. We prove that the empirical distribution of a normalization of λ1, · · · , λm goes to the circular law, that is, the uniform distribution on {z ∈ C; |z| ≤ 1} as m → ∞ with m/n → 0. We also prove that the empirical distribution of λ1, · · · , λm goes to the arc law, that is, the uniform distribution on {z ∈ C; |z| = 1} as m/n → 1. These explain two observations by Życzkowski and Sommers (2000).
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