A nonperturbative theory of randomly branching chains

We present a nonperturbative theory of randomly branching chains (or polymers). The method is based upon the existence of critical points in the large- N limit of O( N ) symmetric vector models. We derive a class of ordinary differential equations governing the behaviour of the partition function with respect to the coupling constants. Solving these equations explicitly, we derive several exact results. Our model is useful to make comparison with the recent nonperturbative studies of two-dimensional quantum gravity since the structure of the differential equations has many parallel features, such as the flow property, the Virasoro structure in the Schwinger-Dyson equation, and so on, with those of random surfaces.