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 x³ᵏ⁻¹e^-x³. Concerning the largest gaps, normalized by n/√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.