Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices

Details Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices

2004-09-28

arxiv.org/abs/math/0409552v2

Mathematics

Probability

Scientific paper

Let $U_m$ be an $m \times m$ Haar unitary matrix and $U_{[m,n]}$ be its $n \times n$ truncation. In this paper the large deviation is proven for the empirical eigenvalue density of $U_{[m,n]}$ as $m/n \to \lambda $ and $n \to \infty$. The rate function and the limit distribution are given explicitly. $U_{[m,n]}$ is the random matrix model of $quq$, where $u$ is a Haar unitary in a finite von Neumann algebra, $q$ is a certain projection and they are free. The limit distribution coincides with the Brown measure of the operator $quq$.

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