Non-perturbative solution of matrix models modified by trace-squared terms

We present a non-perturbative solution of large N matrix models modified by terms of the form g(Trø⁴)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 ∼ e^α+ø. 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^α-ø. 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 gO², 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.