Elementary Proof for Asymptotics of Large Haar-Distributed Unitary Matrices

Details Elementary Proof for Asymptotics of Large Haar-Distributed Unitary Matrices Elementary Proof for Asymptotics of Large Haar-Distributed Unitary Matrices

2007-05-22

arxiv.org/abs/0705.3146v2

Letters in Mathematical Physics 82 (2007) 51-59

Mathematics

Probability

Scientific paper

10.1007/s11005-007-0194-7

We provide an elementary proof for a theorem due to Petz and R\'effy which states that for a random $n\times n$ unitary matrix with distribution given by the Haar measure on the unitary group U(n), the upper left (or any other) $k\times k$ submatrix converges in distribution, after multiplying by a normalization factor $\sqrt{n}$ and as $n\to\infty$, to a matrix of independent complex Gaussian random variables with mean 0 and variance 1.

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