Probability of Initial Ring Closure for Self-Avoiding Walks on the Face-Centered Cubic and Triangular Lattices

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Abstract
The probability of initial ring closure in the self-avoiding walk model of a polymer is investigated. Numerical data on the exact number of self-avoiding walks and polygons on the triangular and face-centered-cubic lattices are presented. It is concluded that the initial ring closure probability for large ring size k varies inversely as kθ with θ≃1 (5/6) in two dimensions and θ=1 (11/12) in three dimensions. It is found empirically that cn, the number of self-avoiding walks of n steps, approximates for large n to A |(jn)| μnwith for the triangular lattice j=−4/3, μ=4.1515, A=1.10, and for the face-centered-cubic lattice j=−7/6, μ=10.035, A=1.04.