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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.

The derivation of series expansions appropriate for low temperatures or high applied magnetic fields for the two-dimensional Ising model of a ferromagnet and antiferromagnet is studied as a temperature grouping. New results are given for the ferromagnetic polynomials for the triangular lattice to order 16, for the ferromagnetic and antiferromagnetic polynomials for the simple quadratic lattice to order 11, and for the honeycomb lattice to order 16.

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