Timothy M Garoni et al 2009 J. Phys. A: Math. Theor. 42 095205 doi:10.1088/1751-8113/42/9/095205
Prudent walks and polygons
Timothy M Garoni, Anthony J Guttmann, Iwan Jensen and John C Dethridge
Show affiliationsWe 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 polygons. In the three-dimensional case we estimate the growth constant for both walks and polygons and also estimate the usual critical exponents γ, ν and α.
- PACS
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
02.10.Ox Combinatorics; graph theory
- MSC
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82B05 Classical equilibrium statistical mechanics (general)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
82B80 Numerical methods (Monte Carlo, series resummation, etc.) (See also 65-XX, 81T80)
82B41 Random walks, random surfaces, lattice animals, etc. (See also 60G50, 82C41)
- Subjects
- Dates
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Issue 9 (6 March 2009)
Received 17 October 2008, in final form 3 January 2009
Published 4 February 2009
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Prudent walks and polygons
Timothy M Garoni et al 2009 J. Phys. A: Math. Theor. 42 095205
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