The Shapes of Random Walks

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Science  24 Jul 1987:
Vol. 237, Issue 4813, pp. 384-389
DOI: 10.1126/science.237.4813.384


A theoretical description of the shape of a random object is presented that is analytically simple in application but quantitatively accurate. The asymmetry of the object is characterized in terms of the invariants of a tensor, analogous to the moment-of-inertia tensor, whose eigenvalues are the squares of the principal radii of gyration. The complications accompanying ensemble averaging because of random processes are greatly reduced when the object is embedded in a space of high dimensionality, d. Exact analytical expressions are presented in the case of infinite spatial dimensions, and a procedure for developing an expansion in powers of l/d is discussed for linear chain and ring-type random walks. The first two terms in such an expansion lead to results for various shape parameters that agree remarkably well with those calculated by computer simulation. The method can be extended to yield an approximate, but extremely accurate, expression for the probability distribution function directly. The theoretical approach discussed here can, in principle, be used to describe the shape of other random fractal objects as well.