  We study the distribution of the minimum spacing between eigenvalues of a
random n by n unitary matrix. The minimum spacing scales as $n^{-4/3}$, not
$n^{-2}$ as would be the case for n independent points on the unit circle,
illustrating the well known phenomenon that the eigenvalues of random matrices
'repel each other'. We derive the distribution for the rescaled minimum spacing
in the limit as $n\to\infty$.
  To find the minimum spacing, we count the number of eigenvalue pairs closer
than $xn^{-4/3}$. We use heuristics to guess that this integer-valued random
variable is Poisson, calculate the actual moments of the limiting distribution,
and find that the actual moments match those of the guess. The matching moments
prove that the heuristic guess is correct, and lead directly to the main
result.
  We prove analogous results for the Gaussian unitary ensemble (GUE) and, with
restrictions, a universal class of unitary ensembles (UUE) studied by Deift,
Kreicherbauer, McLaughlin, Venakides, and Zhou.
