Theory of random walks with limited order of non-self-intersections used to simulate macromolecules

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Abstract
A general theory of random walks subject to partial elimination of self‐intersections has been carried out. Specifically, the theory applies to so‐called self‐avoiding walks of order p, meaning that loops of p or less are forbidden while larger loops are allowed. All walks, used to represent macromolecules, are categorized by the configurations of their last p−1 steps. The theory deals with the successive addition of steps through use of transition matrices, together with their eigenvectors and eigenvalues, taking cognizance of the effect of symmetry. An exact solution is derived in general form and it is found that the mean square end‐to‐end separation of the walks becomes asymptotically linear in n, the number of steps. The slope of the asymptotic line increases with the order p, presumably without limit. All classes of walks, characterized by their different end configurations, asymptotically exhibit similar linear behavior with the same slope but with different intercepts.

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