Statistics of lattice animals (polyominoes) and polygons. (English) Zbl 1009.82010
Summary: 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, \(\tau= 4.062570(8)\), 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 \(\tau> 3.90318\). 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 \(\nu> 0.64115(5)\), 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.
MSC:
82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |
82-04 | Software, source code, etc. for problems pertaining to statistical mechanics |
05B50 | Polyominoes |
Online Encyclopedia of Integer Sequences:
Number of free polyominoes (or square animals) with n cells.Number of fixed polyominoes with n cells.