  This paper discusses Random Matrix Models which exhibit the unusual phenomena
of having multiple solutions at the same point in phase space. These matrix
models have gaps in their spectrum or density of eigenvalues. The free energy
and certain correlation functions of these models show differences for the
different solutions. Here I present evidence for the presence of multiple
solutions both analytically and numerically.
  As an example I discuss the double well matrix model with potential $V(M)=
-{\mu \over 2}M^2+{g \over 4}M^4$ where $M$ is a random $N\times N$ matrix (the
$M^4$ matrix model) as well as the Gaussian Penner model with $V(M)={\mu\over
2}M^2-t \ln M$. First I study what these multiple solutions are in the large
$N$ limit using the recurrence coefficient of the orthogonal polynomials.
Second I discuss these solutions at the non-perturbative level to bring out
some differences between the multiple solutions. I also present the two-point
density-density correlation functions which further characterizes these models
in a new university class. A motivation for this work is that variants of these
models have been conjectured to be models of certain structural glasses in the
high temperature phase.
