  We present a non-perturbative solution of large $N$ matrix models modified by
terms of the form $ g(\Tr\Phi^4)^2$, which add microscopic wormholes to the
random surface geometry. For $g<g_t$ the sum over surfaces is in the same
universality class as the $g=0$ theory, and the string susceptibility exponent
is reproduced by the conventional Liouville interaction $\sim e^{\alpha_+
\phi}$. For $g=g_t$ we find a different universality class, and the string
susceptibility exponent agrees for any genus with Liouville theory where the
interaction term is dressed by the other branch, $e^{\alpha_- \phi}$. This
allows us to define a double-scaling limit of the $g=g_t$ theory. We also
consider matrix models modified by terms of the form $g O^2$, where $O$ is a
scaling operator. A fine-tuning of $g$ produces a change in this operator's
gravitational dimension which is, again, in accord with the change in the
branch of the Liouville dressing.
