Abstract
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 column-convex polygons, as well as partially directed self-avoiding walks), starting with nonlinear functional equations for the generating function. By linearizing these equations, we first give a derivation of the generating functions. The nonlinear equations are further used to compute the thermodynamic critical exponents via a formal perturbation ansatz. Alternatively, taking the continuum limit leads to nonlinear 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 γ u =-1/2, γ i =-1/3, and ϕ=2/3. All models have as their scaling function the logarithmic derivative of the Airy function.
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Prellberg, T., Brak, R. Critical exponents from nonlinear functional equations for partially directed cluster models. J Stat Phys 78, 701–730 (1995). https://doi.org/10.1007/BF02183685
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DOI: https://doi.org/10.1007/BF02183685