Large Deviations of Extreme Eigenvalues of Random Matrices

We calculate analytically the probability of large deviations from its mean of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary, and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an (N) random matrix are positive (negative) decreases for large N as - (0) N^2 where the parameter characterizes the ensemble and the exponent (0) = (ln3) /4=0. 274 653 is universal. We also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number, thus generalizing the celebrated Wigner semicircle law. The density of states generically exhibits an inverse square-root singularity at.