0 Statistics of lattice animals ( polyominoes ) and polygons

We have developed an improved algorithm that allows us to enumerate the number of site animals (polyominoes) on the square lattice up to size 46. Analysis of the resulting series yields an improved estimate, τ = 4.062570(8), for the growth constant of lattice animals and confirms to a very high degree of certainty that the generating function has a logarithmic divergence. We prove the bound τ > 3.90318. We also calculate the radius of gyration of both lattice animals and polygons enumerated by area. The analysis of the radius of gyration series yields the estimate ν = 0.64115(5), for both animals and polygons enumerated by area. The mean perimeter of polygons of area n is also calculated. A number of new amplitude estimates are given. The enumeration of lattice animals is a classical combinatorial problem of great interest both intrinsically and as a paradigm of recreational mathematics [1]. A lattice animal is a finite set of nearest neighbour sites on a lattice. The fundamental problem is the calculation of the number of animals, bn, containing n sites. In the physics literature lattice animals are very often called clusters due to their very close relationship to percolation problems [2]. Series expansions for various percolation properties, such as the percolation probability or the average cluster size, can be obtained from the perimeter polynomials. These in turn can be calculated by counting the number of lattice animals bn,m according to their size n and perimeter m [3, 4]. Lattice animals have also been suggested as a model of branched polymers [5]. In mathematics, and combinatorics in particular, the term polyominoes is frequently used. A polyomino is a set of lattice cells joined at their edges. So polyominoes are identical to site animals on the dual lattice. Furthermore, the enumeration of lattice animals has traditionally served as a benchmark for computer performance and algorithm design [6]–[12]. The enumeration of self-avoiding polygons is another classical combinatorial problem [13]. Most attention has been paid to the enumeration by perimeter, but enumeration by area is an equally interesting problem. (For polyominoes, the ordinary generating function of the number of polyominoes of perimeter n has zero radius of convergence [14] and hence is of rather less interest). Polygons enumerated by area are just the “hole-free” subset of polyominoes. There are exponentially fewer polygons than polyominoes [15], but on universality grounds one would e-mail: [email protected] e-mail: [email protected]