Endless self-avoiding walks

Abstract

We introduce a self-avoiding walk model for which end-effects are completely eliminated. We enumerate the number of these walks for various lattices in dimensions two and three, and use these enumerations to study the properties of this model. We find that endless self-avoiding walks have the same connective constant as self-avoiding walks, and the same Flory exponent $\nu$. However, there is no power law correction to the exponential number growth for this new model, i.e. the critical exponent $\gamma =1$ exactly. In addition, we have convincing numerical evidence to support the hypothesis that the amplitude for the number growth is a universal quantity, and somewhat weaker evidence which suggests that the number growth has no analytic corrections to scaling. The technique by which end-effects are eliminated may be generalised to other models of polymers such as interacting self-avoiding walks.

26 pages, 19 figures; typos fixed, expanded arguments for $\gamma$ and $\nu$, added explanation for absence of analytic corrections to scaling, changed conclusion about existence of anti-ferromagnetic singularity, and added an example of a knotted endless self-avoiding walk

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Cited by in corpus (3)

• Theta-point polymers in the plane and Schramm-Loewner evolution
• Monte Carlo Simulation of Long Hard-Sphere Polymer Chains in Two to Five Dimensions
• Bounded affine permutations II. Avoidance of decreasing patterns