  In this paper we study the limiting distribution of the $k$ smallest gaps
between eigenvalues of three kinds of random matrices -- the Ginibre ensemble,
the Wishart ensemble and the universal unitary ensemble. All of them follow a
Poissonian ansatz. More precisely, for the Ginibre ensemble we have a global
result in which the $k$-th smallest gap has typical length $n^{-3/4}$ with
density $x^{4k-1}e^{-x^4}$ after normalization. For the Wishart and the
universal unitary ensemble, it has typical length $n^{-4/3}$ and has density
$x^{3k-1}e^{-x^3}$ after normalization.
