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How a random walk covers a finite lattice

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  • Brummelhuis, M.J.A.M.
  • Hilhorst, H.J.

Abstract

A random walker is confined to a finite periodic d-dimensional lattice of N initially white sites. When visited by the walk a site is colored black. After t steps of the walk, for t scaled appropriately with N, we determined the structure of the set of white sites. The variance of their number has a line of critical points in the td plane, which separates a mean-field region from a region with enhanced fluctuations. At d = 2 the critical point becomes a critical interval. Moreover, for d = 2 the set of white sites is fractal with a fractal dimensionality whose t-dependence we determine.

Suggested Citation

  • Brummelhuis, M.J.A.M. & Hilhorst, H.J., 1992. "How a random walk covers a finite lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 185(1), pages 35-44.
    • Handle: RePEc:eee:phsmap:v:185:y:1992:i:1:p:35-44
    DOI: 10.1016/0378-4371(92)90435-S
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    Cited by:

    1. Ney-Nifle, M. & Hilhorst, H.J., 1993. "Chaos exponents in spin glasses," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 193(1), pages 48-78.
    2. Cieplak, Marek & Banavar, Jayanth R., 1993. "Scaling and phase transitions in random systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 194(1), pages 63-71.
    3. Ney-Nifle, M. & Hilhorst, H.J., 1993. "Renormalization theory and chaos exponents in random systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 194(1), pages 462-470.

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