  We consider random topologies of surfaces generated by cubic interactions.
Such surfaces arise in various contexts in 2-dimensional quantum gravity and as
world-sheets in string theory. Our results are most conveniently expressed in
terms of a parameter h = n/2 + \chi, where n is the number of interaction
vertices and \chi is the Euler characteristic of the surface. Simulations and
results for similar models suggest that Ex[h] = log (3n) + \gamma + O(1/n) and
Var[h] = log (3n) + \gamma - \pi^2/6 + O(1/n). We prove rigourously that Ex[h]
= log n + O(1) and Var[h] = O(log n). We also derive results concerning a
number of other characteristics of the topology of these random surfaces.
