Extreme Gaps Between Eigenvalues Of Random Matrices

This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian unitary ensemble. In particular, the kth smallest gap, normalized by a factor n(-4/3), has a limiting density proportional to x(3k-1)e(-x3). Concerning the largest gaps, normalized by n/root log n, they converge in L-p to a constant for all p > 0. These results are compared with the extreme gaps between zeros of the Riemann zeta function.