Scaling of the atmosphere of self-avoiding walks
AL Owczarek, T Prellberg
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL | IOP PUBLISHING LTD | Published : 2008
The 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. © 2008 IOP Publishing Ltd.