Nathan Clisby et al 2007 J. Phys. A: Math. Theor. 40 10973 doi:10.1088/1751-8113/40/36/003
Nathan Clisby1, Richard Liang2 and Gordon Slade3
Show affiliationsWe introduce a new method for the enumeration of self-avoiding walks based on the lace expansion. We also introduce an algorithmic improvement, called the two-step method, for self-avoiding walk enumeration problems. We obtain significant extensions of existing series on the cubic and hypercubic lattices in all dimensions d ≥ 3: we enumerate 32-step self-avoiding polygons in d = 3, 26-step self-avoiding polygons in d = 4, 30-step self-avoiding walks in d = 3, and 24-step self-avoiding walks and polygons in all dimensions d ≥ 4. We analyze these series to obtain estimates for the connective constant and various critical exponents and amplitudes in dimensions 3 ≤ d ≤ 8. We also provide major extensions of 1/d expansions for the connective constant and for two critical amplitudes.
05.40.Fb Random walks and Levy flights
05.70.Jk Critical point phenomena
05.10.-a Computational methods in statistical physics and nonlinear dynamics
82B41 Random walks, random surfaces, lattice animals, etc. (See also 60G50, 82C41)
05A15 Exact enumeration problems, generating functions (See also 33Cxx, 33Dxx)
Issue 36 (7 September 2007)
Received 21 May 2007
Published 21 August 2007
Nathan Clisby et al 2007 J. Phys. A: Math. Theor. 40 10973
Francisco J Herranz and Mariano Santander 2002 J. Phys. A: Math. Gen. 35 6601
Duncan A Brown (for the LIGO Scientific Collaboration) 2005 Class. Quantum Grav. 22 S1097
A N F Aleixo and A B Balantekin 2007 J. Phys. A: Math. Theor. 40 3915
Pavel Kurasov and Marlena Nowaczyk 2005 J. Phys. A: Math. Gen. 38 4901
John T Whelan et al 2005 Class. Quantum Grav. 22 S1087
J F Stephany 1979 J. Phys. A: Math. Gen. 12 1667
Björn Poppe et al 2007 Phys. Med. Biol. 52 2921
Katsuaki Asano et al. 2009 ApJ 699 953
Iwan Jensen 2004 J. Phys. A: Math. Gen. 37 6899