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 $n^{-4\mathrm{/}3}$, has a limiting density proportional to $x^{3k-1}{e}^{-x^3}$. Concerning the largest gaps, normalized by $n\mathrm{/}\sqrt{\log n}$, they converge in ${\mathrm{L}}^p$ to a constant for all $p>0$. These results are compared with the extreme gaps between zeros of the Riemann zeta function.