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 N₏ branched polymers containing N₁ monomers, N₅f-functional units, and N₋ 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 (N₁) of animals per site containing N₁ bonds grows as {N₁}^-^Nb as N₁ tends to infinity, leading to a singularity in the generating function (K) ₍₁K^NbA (N₁) of the form (K) |K-{K₂|}^-1, where K₂=^-1. Mean-field theory for this and other animal functions is valid above an upper critical dimension d₂ of 8. For the related polymer problem, d₂ is 8 in good solvents and 6 in 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 and obtain their usual mean-field values of 1 and for d>d₂. First-order corrections in d₂-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 =52, ₀=12, and ₀=14 for d>d₂; =52 agrees with exact calculations on a Cayley tree and ₀=14 with calculations by Zimm and Stackmayer of the radius of gyration of a branched polymer. For d<d₂, -1= (3-2{₃) } (2-{₃) }, ₀= (2-{₃) }, and ₀= (2-{₃) }, where ₃ is a critical exponent controlling the critical behavior of an irrelevant three-point vertex that is of order d-d₂ for d near d₂. Both the constant-order-parameter and constant-fugacity theories violate the hyperscaling d=2-, where is the specific-heat exponent.