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

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)| μn with 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.