Prudent walks and polygons
Timothy M Garoni, Anthony J Guttmann, Iwan Jensen, John C Dethridge
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL | IOP PUBLISHING LTD | Published : 2009
We have produced extended series for two-dimensional prudent polygons, based on a transfer matrix algorithm of complexity O(n5), for a series of n-step polygons. For prudent polygons in two dimensions we find the growth constant to be smaller than that for the corresponding walks, and by considering three distinct subclasses of prudent walks and polygons, we find that the growth constant for polygons varies with class, while for walks it does not. We give exact values for the critical exponents γ and α for walks and polygons, respectively. We have extended the definition of prudent walks to three dimensions and produced series expansions, using a back-tracking algorithm, for both walks and p..View full abstract
We would like to thank Simone Rinaldi and Enrica Duchi for introducing us to this problem, and Mireille Bousquet-Melou for several enlightening discussions and access to her results prior to publication. Similarly, we thank Uwe Schwerdtfeger for provision of his results prior to publication. We are also grateful to Andrew Conway, who checked much of the data independently, and to the referees whose queries resulted in a significantly improved manuscript. The calculations presented in this paper were performed on the facilities of the Australian Partnership for Advanced Computing (APAC) and the Victorian Partnership for Advanced Computing (VPAC). We gratefully acknowledge financial support from the Australian Research Council.