A L Owczarek and T Prellberg 2008 J. Phys. A: Math. Theor. 41 375004 doi:10.1088/1751-8113/41/37/375004
A L Owczarek1 and T Prellberg2
Show affiliationsThe number of free sites next to the end of a self-avoiding walk is known as the atmosphere of the walk. The average atmosphere can be related to the number of configurations. Here we study the distribution of atmospheres as a function of length and how the number of walks of fixed atmosphere scale. Certain bounds on these numbers can be proved. We use Monte Carlo estimates to verify our conjectures in two dimensions. Of particular interest are walks that have zero atmosphere, which are known as trapped. We demonstrate that these walks scale in the same way as the full set of self-avoiding walks, barring an overall constant factor.
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
82B80 Numerical methods (Monte Carlo, series resummation, etc.) (See also 65-XX, 81T80)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
82B41 Random walks, random surfaces, lattice animals, etc. (See also 60G50, 82C41)
Issue 37 (19 September 2008)
Received 6 June 2008, in final form 22 July 2008
Published 13 August 2008
A L Owczarek and T Prellberg 2008 J. Phys. A: Math. Theor. 41 375004
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