The asymptotic form of the cluster partition function in a two-dimensional lattice gas

The partition function, Zp, of a cluster of p particles in a lattice gas depends on the number of lattice embeddings of labelled, connected graphs with p points and a given number of lines. We have determined the asymptotic behavior of this quantity for the triangular lattice. It appears that a similar behavior obtains in the square lattice. We find that Zp⋍ekp−μ√p as p→∞. For small clusters, the surface energy is significantly greater than its asymptotic value.