Statistics of trees and branched polymers from a generalised Hilhorst model

Abstract

A generalisation of the Hilhorst model in which at each site, x, on a lattice, there is an n-state variable nu (x), and an s-state variable, sigma (x), which interact via a Hamiltonian H=-nK Sigma _(x,x') delta ^{nu (x) nu (x')}(s delta ^{sigma (x) sigma (x')}-1)-h Sigma _x delta ^{nu (x)1}-1) is introduced. It is shown that if (s-1)= lambda n, the n=0 limit of the partition function for this model is the generating function for trees in which ln K is the chemical potential for bonds (monomers), ln lambda for the number of trees (polymers) an ln h for the number of free ends of all trees. Fields which mark any point and fields which mark only external points of a polymer are identified. The above Hamiltonian is converted to a field theory which is used to discuss the dependence on the monomer number, N, of critical properties such as the radius of gyration of branched polymers with a small number of branchings. It is shown that these properties are controlled by the usual n=0 polymer fixed point.