  We introduce a family of rotationally invariant random matrix ensembles
characterized by a parameter $\lambda$. While $\lambda=1$ corresponds to
well-known critical ensembles, we show that $\lambda \ne 1$ describes "L\'evy
like" ensembles, characterized by power law eigenvalue densities. For $\lambda
> 1$ the density is bounded, as in Gaussian ensembles, but $\lambda <1$
describes ensembles characterized by densities with long tails. In particular,
the model allows us to evaluate, in terms of a novel family of orthogonal
polynomials, the eigenvalue correlations for L\'evy like ensembles. These
correlations differ qualitatively from those in either the Gaussian or the
critical ensembles.
