Statistics of lattice animals and dilute branched polymers

A field theory, recently introduced by the authors, whose partition function yields the generating function for the number of configurations of Nₚ branched polymers containing Nb monomers, Nff-functional units, and NL loops, is used to study the statistics of animals (clusters on a lattice) and the intimately related problem of the statistics of branched polymers in the dilute limit. The number A(Nb) of animals per site containing Nb bonds grows as Nb^-θλNb as Nb tends to infinity, leading to a singularity in the generating function Ξ(K)ΣNbKNbA(Nb) of the form Ξ(K)~|K-Kc|^θ-1, where Kc=λ^-1. Mean-field theory for this and other animal functions is valid above an upper critical dimension dc of 8. For the related polymer problem, dc is 8 in good solvents and 6 in θ solvents. In both cases, the critical behavior of generating and correlation functions depends on the nature of applied constraints. If a special combination of fields corresponding to the natural order parameter of the field theory is held constant, the susceptibility and correlation-length exponents γ and ν obtain their usual mean-field values of 1 and {} for d>dc. First-order corrections in dc-d are calculated for these and other exponents. If fugacities are held constant, there is a Fisher-like renormalization of critical exponents leading to animal's exponents θ=5/2, γₐ=1/2, and νₐ=1/4 for d>dc; θ=5/2 agrees with exact calculations on a Cayley tree and νₐ=1/4 with calculations by Zimm and Stackmayer of the radius of gyration of a branched polymer. For d<dc, θ-1=(3γ-2μ₃)(2γ-μ₃), γₐ=γ(2γ-μ₃), and νₐ=ν(2γ-μ₃), where μ₃ is a critical exponent controlling the critical behavior of an irrelevant three-point vertex that is of order d-dc for d near dc. Both the constant-order-parameter and constant-fugacity theories violate the hyperscaling dν=2-α, where α is the specific-heat exponent.