CRITICAL EXPONENTS FROM NONLINEAR FUNCTIONAL-EQUATIONS FOR PARTIALLY DIRECTED CLUSTER-MODELS

We present a method for the derivation of the generating function and computation of critical exponents for several cluster models (staircas e, bar-graph, and directed column-convex polygons, as well as partiall y directed self-avoiding walks), starting with nonlinear functional eq uations for the generating function. By linearizing these equations, w e first give a derivation of the generating functions. The nonlinear e quations are further used to compute the thermodynamic critical expone nts via a formal perturbation ansatz. Alternatively, taking the contin uum limit leads to nonlinear differential equations, from which one ca n extract the scaling function. We find that all the above models are in the same universality class with exponents gamma(u) = -1/2, gamma(t ) = -1/3, and phi = 2/3. All models have as their scaling function the logarithmic derivative of the Airy function.