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By A L Owczarek and T Prellberg

Abstract

The number of free sites next to the end of a self-avoiding walk is known as the atmosphere. 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. 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. 1 Consider an n-step self-avoiding walk ω = (ω0, ω1,...,ωn) with n + 1 sites ωi ∈ Z d for d ≥ 2, and steps having unit length, i.e. |ωi+1 − ωi | = 1. Figure 1: A self-avoiding walk on the square lattice Z 2 with n = 20 steps and an atmosphere a = 2. The number of edges that can be appended to the last visited vertex ωn to create an (n + 1)-step is called the atmosphere of the walk ω. Clearly the smallest value of the atmosphere is zero, in which case the walk is called trapped. A zero-step self-avoiding walk has atmosphere 2d, and for n ≥ 1 any n-step self-avoiding walk has atmosphere of at most 2d − 1. We partition the set of n-step self-avoiding walks by the value of their atmosphere. Denote by cn the number of n-step self-avoiding walks starting at ω0 = 0, and by c (a) n number of n-step self-avoiding walks starting at ω0 = 0 with atmosphere a. The subject of this paper is the fraction of n-step self-avoiding walks with fixed atmosphere, and its limiting behaviour as n → ∞. p (a) n = c(a) n cn the, (1) The atmosphere was introduced by Rechnitzer and Janse van Rensburg [1] for selfavoiding walks. Since it is well established theoretically, though not proven, that cn ∼ A1µ n n γ−1 (2) they immediately pointed out that the mean atmosphere could be used to estimate the connective constant µ and exponent γ sinc

Year: 2008
OAI identifier: oai:CiteSeerX.psu:10.1.1.311.354
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