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Square Lattice Self-Avoiding Walks and Polygons
A.J. Guttmann1 and A.R. Conway2
1Department of Mathematics and Statistics, The University of Melbourne, Parkville, Victoria 3010, Australia
tonyg@ms.unimelb.edu.au
2Silicon Genetics, 2601 Spring Street, Redwood City, CA 94063, USA
Annals of Combinatorics 5 (3) p.319-345 September, 2001
AMS Subject Classification: 05A15
Abstract:
We give an algorithm for the enumeration of self-avoiding walks on the (anisotropic) square lattice. Application of the algorithm on a 1024 processor Intel Paragon supercomputer resulted in a 51 term series. For (isotropic) square lattice self-avoiding polygons, a related algorithm has produced a 90 term series. Careful analysis provides compelling evidence for simple rational values of the exponents in both the dominant and subdominant terms in the asymptotic form of the coefficients. We also advance compelling arguments --- but not a proof --- that the generating function for SAW is not differentiably finite. The corresponding result for SAP has recently been proved.
Keywords: enumeration, self-avoiding walks, polygons

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