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

Let U_m 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.