Distribution of the number of distinct sites visited by random walks in disordered lattices

Lazaros K. Gallos; Panos Argyrakis; Klaus W. Kehr

doi:10.1103/PhysRevE.52.1520

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Distribution of the number of distinct sites visited by random walks in disordered lattices

Lazaros K. Gallos and Panos Argyrakis

Klaus W. Kehr

Department of Physics, University of Thessaloniki, GR-54006 Thessaloniki, Greece

Institut für Festkörperforschung, Forschungszentrum Jülich GmbH, D-52425 Jülich, Germany

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Phys. Rev. E 52, 1520 – Published 1 August, 1995

DOI: https://doi.org/10.1103/PhysRevE.52.1520

Abstract

The distributions of the number of distinct sites $S_{n}$ visited by random walks of n steps in the infinite cluster of two-dimensional lattices at the percolation threshold are studied. Different lattice sizes, different origins of the walks, and different realizations of the disorder are investigated by Monte Carlo simulations. The distribution of the mean values of 〈 $S_{n}$ 〉 appears to have self-averaging features. The probability distribution of the normalized values of 〈 $S_{n}$ 〉 is investigated with respect to its multifractal behavior. The distributions of the probabilities p( $S_{n}$ ) for fixed $S_{n}$ are presented and analyzed. These distributions are wide and their moments show behavior that cannot be characterized by multifractal scaling exponents.

References (13)

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P. Argyrakis and K.W. Kehr (unpublished).

[c1] A. Bunde and S. Havlin, Fractals and Disordered Systems (Springer, Berlin, 1991).

[c2] For a review, see G. Paladin and A. Vulpiani, Phys. Rep. 156, 147 (1987).

[c3] A. Bunde, S. Havlin and H.E. Roman, Phys. Rev. A 42, 6274 (1990); E.Eisenberg, Phys. Rev. E 47, 2333 (1993).

[c4] Y. Gefen, A. Aharony and S. Alexander, Phys. Rev. Lett. 50, 1399 (1983).

[c5] I. Webman, Phys. Rev. Lett 52, 220 (1984).

[c6] T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia and B.I. Shraiman, Phys. Rev. A 33, 1141 (1986).

[c7] B.B. Mandelbrot, J. Fluid Mech. 62, 331 (1974).

[c8] P. Grassberger, Phys. Lett. A 97, 227 (1983).

[c9] K.P.N. Murthy, S. Rajasekar and K.W. Kehr, J. Phys. A 27, L107 (1994).

[c10] See, e.g., D. Stauffer and A.Aharony, Introduction to Percolation Theory, 2nd edition (Taylor and Francis, London, 1991).

[c11] J. Hoshen and R. Kopelman, Phys. Rev. B 14, 3438 (1976).

[c12] J. Feder, Fractals (Plenum, New York, 1988).

[c13] P. Argyrakis and K.W. Kehr (unpublished).

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Distribution of the number of distinct sites visited by random walks in disordered lattices

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