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(TrΦ4)2, which add microscopic wormholes to the random surface geometry. For g < gt 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 ∼ eα+φ. For g = gt 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, eα−φ. This allows us to define a double-scaling limit of the g = gt 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.