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-Nb as Nb tends to infinity, leading to a singularity in the generating function (K)NbKNbA(Nb) of the form (K) K-Kc -1, where Kc=-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 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>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=52, a=12, and a=14 for d>dc;=52 agrees with exact calculations on a Cayley tree and a=14 with calculations by Zimm and Stackmayer of the radius of gyration of a branched polymer. For d<dc,-1=(3-2 3)(2-3), a=(2- 3), and=(2-3), where 3 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 d=2-, where is the specific-heat exponent. © 1979 The American Physical Society.