Statistical Computation of Mean Dimensions of Polymer Molecules. IV

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Abstract
In a further investigation of the nonintersecting random walk problem, two new types of random walks have been examined to determine the effect of the magnitude of the ``excluded volume'' on the limiting behavior of the ratio, 〈rn2〉n, where 〈rn2〉 is the mean square length of nonintersecting random walks of n steps. In two dimensions, 〈rn2〉/n normally diverges, but for walks in a right‐left two‐choice 90° lattice, for which exclusions are counted on odd steps only, the ``excluded area'' is reduced to such an extent that convergence of 〈rn2〉Av/n may occur. In three dimensions, 〈rn2〉/n may converge, but if random walks in a three‐choice tetrahedral lattice are generated subject to the condition that walks cannot return to lattice points within unit distances from previously occupied sites, the ``excluded volume'' is sufficiently increased so that 〈rn2〉/n certainly diverges as n→∞. In addition to the above, our process for selecting the random numbers used to generate random walks has been re‐examined more fully and has been found to be satisfactory. We have also examined values for mean square lengths of failures which might result from backward steps. These distances agree with an empirical relationship found to hold for the other types of failures, thus providing further justification for an earlier theoretical interpretation of the nonintersecting random walk problem.

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