Self-Avoiding Random Walks. I. Simple Properties of Intermediate-Length Walks

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Abstract
A new method is presented for the rapid detection of self‐intersections in random walks generated by Monte Carlo processes. Reduction in required computer time made possible by this method allows the generation of walks containing significantly more steps than previously possible, thus corresponding to more realistic linear macromolecular dimensions. Results obtained by this method verify, for the tetrahedral lattice, a conjecture of Domb's that 〈Rn2〉, the mean‐square end‐to‐end distance, increases essentially as the 6/5 power of n, the number of steps in a walk. In addition, a new parameter 〈xn〉 the average maximum x‐axis extension of a walk, is calculated. The square of this parameter is found to have essentially the same exponential step dependence as 〈Rn2〉 and may be calculated more precisely than it. Further, 〈Dn2〉, the average maximum squared diameter of a walk estimated from 〈xn〉, having the property that essentially all of an average self‐avoiding walk of n steps is contained within a sphere of squared diameter 〈Dn2〉, is shown to be approximately 2.2 times 〈Rn2〉. The results presented are valid to 1700 steps.

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