A large deviation theorem for the empirical eigenvalue distribution of random unitary matrices

It is Shown that the empirical eigenvalue distribution of suitably distribu ted random unitary matrices satisfies the large deviation principle as the matrix size goes to infinity. The primary term of the rate function is the logarithmic energy (or the minus sign of Voiculescu's free entropy). Exampl es of random unitaries are also discussed, one of them is related to the wo rk of Gross and Witten in quantum physics. (C) 2000 Editions scientifiques et medicales Elsevier SAS.