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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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D. Dürr, S. Goldstein, N. Zanghì: Quantum equilibrium and the origin of absolute uncertainty. J. Statist. Phys
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2004
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S. Goldstein, J.L. Lebowitz, C. Mastrodonato, R. Tumulka, N.Zanghì: Typicality of the GAP Measure. In preparation
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S. Goldstein, J.L. Lebowitz, R. Tumulka, N. Zanghì: On the Distribution of the Wave Function for Systems in Thermal Equilibrium. J. Statist. Phys
2006
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