Statistics of lattice animals (polyominoes) and polygons

Abstract

We have developed an improved algorithm that allows us to enumerate the\nnumber of site animals (polyominoes) on the square lattice up to size 46.\nAnalysis of the resulting series yields an improved estimate, $\\tau =\n4.062570(8)$, for the growth constant of lattice animals and confirms to a very\nhigh degree of certainty that the generating function has a logarithmic\ndivergence. We prove the bound $\\tau > 3.90318.$ We also calculate the radius\nof gyration of both lattice animals and polygons enumerated by area. The\nanalysis of the radius of gyration series yields the estimate $\\nu =\n0.64115(5)$, for both animals and polygons enumerated by area. The mean\nperimeter of polygons of area $n$ is also calculated. A number of new amplitude\nestimates are given.\n