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An Infinite Number of Effectively Infinite Clusters in Critical Percolation

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Abstract

An infinite number of effectively infinite clusters are predicted at the percolation threshold, if “effectively infinite” means that a cluster's mass increases with a positive power of the lattice size L. All these cluster masses increase as L D with the fractal dimension D = d − β/v, while the mass of the rth largest cluster for fixed L decreases as 1/r λ, with λ = D/d in d dimensions. These predictions are confirmed by computer simulations for the square lattice, where D = 91/48 and λ = 91/96.

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Authors and Affiliations

  1. Physics Department, St. Francis Xavier University, Antogonish, Box 5000, Nova Scotia, Canada, B2G 2W5

    Naeem Jan

  2. School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

    Dietrich Stauffer & Amnon Aharony

  3. Institute for Theoretical Physics, Cologne University, 50923, Cologne, Germany

    Dietrich Stauffer

Authors
  1. Naeem Jan
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  2. Dietrich Stauffer
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  3. Amnon Aharony
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Jan, N., Stauffer, D. & Aharony, A. An Infinite Number of Effectively Infinite Clusters in Critical Percolation. Journal of Statistical Physics 92, 325–330 (1998). https://doi.org/10.1023/A:1023008021962

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