Random walks with intersections: Static and dynamic fractal properties

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Abstract
Static and dynamic properties of the fractal sets generated by free and k-tolerant walks are analyzed in detail. A rather complete picture is obtained for the set of the intersections of free random walks, using correlation functions like the probability of visiting one or more sites m times. Monte Carlo enumerations, jointly with rather sophisticated numerical analysis, are used to determine the fractal dimension of the set of the self-intersections of k-tolerant walks. The results are used to throw new light on the Flory argument for polymer chains with excluded-volume effects; the universal behavior of k-tolerant walks is explained in a coarse-grained reinterpretation of the Flory approximation. Diffusion on the same class of walks allows us to discuss also their universality with respect to dynamical properties. In particular, a spectral dimension equal to (4/3 is obtained for the free random walk in d=2 dimensions.

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