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

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Let Uₘ be an m × m Haar unitary matrix and U[ₘ,ₙ] be its n × n truncation. In this paper the large deviation is proven for the empirical eigenvalue density of U[ₘ,ₙ] as m/n → λ and n → ∞. 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.