ADSORPTION OF UNIFORM LATTICE ANIMALS WITH SPECIFIED TOPOLOGY

We study the effects of topology on the free energy of uniform lattice animals interacting with a surface. Topology is specified by fixing an abstract graph, tau, and the lattice animals considered are embeddings of tau in the square and simple cubic lattice. We prove that such embeddings in the simple cubic lattice and interacting with a plane have the same free energy as self-avoiding walks independent of the choice of tau and independent of whether or not the embeddings are restricted to be uniform. For embeddings in the square lattice and interacting with a line, we prove that the free energy may differ from that for walks depending on whether tau has a cut edge. Further restricting the embeddings to be uniform forces the free energy to be different from that for walks for all tau (except the tau corresponding to walks) and we obtain bounds on the free energy which depend on the number of branches, cycles and vertices of degree 3 and 4 in the graph.