Lattice statistics of branched polymers with specified topologies

Abstract

The authors study the numbers of lattice animals with specified topologies. They prove that the growth constant for animals with c cycles and n_k vertices of degree k (k=3, . . ., 2d), weakly embeddable in a d-dimensional hypercubic lattice, equal to mu , the growth constant for self-avoiding walks. For some particular topologies they derive upper and lower bounds for the corresponding critical exponents and estimate the values of these exponents by deriving and analysing new exact enumeration data. They conjecture that their previous result for trees with b branches (that the exponent is gamma +b-1, where gamma is the self-avoiding walk exponent) is also valid in the more general case in which the cyclomatic index (c) is non-zero; i.e. for b>or=1, the exponent does not depend on c. They show that their results are consistent with renormalisation group arguments that the universality class of branched polymers is independent of cycle and (non-zero) branching fugacities.