  Scaling level-spacing distribution functions in the ``bulk of the spectrum''
in random matrix models of $N\times N$ hermitian matrices and then going to the
limit $N\to\infty$, leads to the Fredholm determinant of the sine kernel
$\sin\pi(x-y)/\pi (x-y)$. Similarly a double scaling limit at the ``edge of the
spectrum'' leads to the Airy kernel $[{\rm Ai}(x) {\rm Ai}'(y) -{\rm Ai}'(x)
{\rm Ai}(y)]/(x-y)$. We announce analogies for this Airy kernel of the
following properties of the sine kernel: the completely integrable system of
P.D.E.'s found by Jimbo, Miwa, M{\^o}ri and Sato; the expression, in the case
of a single interval, of the Fredholm determinant in terms of a Painlev{\'e}
transcendent; the existence of a commuting differential operator; and the fact
that this operator can be used in the derivation of asymptotics, for general
$n$, of the probability that an interval contains precisely $n$ eigenvalues.
