Critical Exponents from Non-Linear 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 (staircase, bar-graph, and directed columnconvex polygons, as well as partially directed self-avoiding walks), starting with nonlinear functional equations for the generating function. By linearising these equations, we first give a derivation of the generating functions. The non-linear equations are further used to compute the thermodynamic critical exponents via a formal perturbation ansatz. Alternatively, taking the continuum limit leads to non-linear differential equations, from which one can extract the scaling function. We find that all the above models are in the same universality class with exponents: γu = −1/2, γt = −1/3 and φ = 2/3. All models have as their scaling function the logarithmic derivative of the Airy function. PACS numbers: 05.50.+q, 05.70.fh, 61.41.+e