Abstract
We have developed an improved algorithm that allows us to enumerate the number of site animals on the square lattice up to size 46. We also calculate the number of lattice trees up to size 44 and the radius of gyration of both lattice animals and trees up to size 42. Analysis of the resulting series yields an improved estimate, λ=4.062570(8), for the growth constant of lattice animals, and, λ0=3.795254(8), for the growth constant of trees, and confirms to a very high degree of certainty that both the animal and tree generating functions have a logarithmic divergence. Analysis of the radius of gyration series yields the estimate, ν=0.64115(5), for the size exponent.
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REFERENCES
S. Golomb, Polyominoes: Puzzles, Patterns, Problems and Packings, 2nd ed. (Princeton U.P, Princeton, New Jersey, 1994).
D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd ed. (Taylor & Francis, London, 1992).
C. Domb, Nature 184:509 (1959).
M. F. Sykes and M. Glen, J. Phys. A 9:87 (1976).
W. F. Lunnon, in Computers in Number Theory, A. O. L. Atkin and B. J. Birch, eds. (Academic Press, London, 1971).
J. L. Martin, in Phase Transitions and Critical Phenomena, Vol. 3, C. Domb and M. S. Green, eds. (Academic Press, London, 1974).
D. H. Redelmeier, Disc. Math. 36:191 (1981).
S. Redner, J. Stat. Phys. 29:309 (1982).
J. L. Martin, J. Stat. Phys. 58:749 (1990).
S. Mertens, J. Stat. Phys. 58:1095 (1990).
S. Mertens and M. E. Lautenbacher, J. Stat. Phys. 66:669 (1992).
A. R. Conway, J. Phys. A. 28:335 (1995).
T. Lubensky and J. Isaacson, Phys. Rev. A 20:2130 (1979).
J. A. M. S. Duarte and H. J. Ruskin, J. Phys. (Paris) 42:1588 (1981).
A. R. Conway and A. J. Guttmann, J. Phys. A. 28:891 (1995).
T. Oliveira e Silva, http://www.inesca.pt/~tos/animals.html.
I. G. Enting, J. Phys. A. 13:3713 (1980).
B. Derrida and L. De Seze, J. Physique 43:475 (1982).
K. Mehlhorn, Data Structures and Algorithms I: Sorting and Searching, EATCS Monographs on Theoretical Computer Science (Springer-Verlag, Berlin, 1984).
D. E. Knuth, Seminumerical Algorithms (The Art of Computer Programming 2) (Addison_Wesley, Reading, Massachusetts, 1969).
I. Jensen and A. J. Guttmann, J. Phys. A 33:L257 (2000).
A. J. Guttmann, in Phase Transitions and Critical Phenomena, Vol. 13, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1989).
S. You and E. J. Janse van Rensburg, Phys. Rev. E 58:3971 (1998).
B. M. I. Rands and D. J. A. Welsh, IAM J. Appl. Math. 27:1 (1981).
S. G. Whittington and C. E. Soteros, in Disorder in Physical Systems, G. R. Grimmett and D. J. A. Welsh, eds. (Clarendon Press, Oxford, 1990).
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Jensen, I. Enumerations of Lattice Animals and Trees. Journal of Statistical Physics 102, 865–881 (2001). https://doi.org/10.1023/A:1004855020556
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DOI: https://doi.org/10.1023/A:1004855020556