Number of spanning clusters at the high-dimensional percolation thresholds

Santo Fortunato, Amnon Aharony, Antonio Coniglio, and Dietrich Stauffer
Phys. Rev. E 70, 056116 – Published 18 November 2004

Abstract

A scaling theory is used to derive the dependence of the average number k of spanning clusters at threshold on the lattice size L. This number should become independent of L for dimensions d<6 and vary as lnL at d=6. The predictions for d>6 depend on the boundary conditions, and the results there may vary between Ld6 and L0. While simulations in six dimensions are consistent with this prediction [after including corrections of order ln(lnL)], in five dimensions the average number of spanning clusters still increases as lnL even up to L=201. However, the histogram P(k) of the spanning cluster multiplicity does scale as a function of kX(L), with X(L)=1+constL, indicating that for sufficiently large L the average k will approach a finite value: a fit of the five-dimensional multiplicity data with a constant plus a simple linear correction to scaling reproduces the data very well. Numerical simulations for d>6 and for d=4 are also presented.

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  • Received 11 July 2004

DOI:https://doi.org/10.1103/PhysRevE.70.056116

©2004 American Physical Society

Authors & Affiliations

Santo Fortunato

  • Fakultät für Physik, Universität Bielefeld, D-33501 Bielefeld, Germany

Amnon Aharony

  • School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel and Department of Physics, Ben Gurion University, Beer Sheva 84105, Israel

Antonio Coniglio

  • Dipartimento di Scienze Fisiche, Università di Napoli “Federico II” and Unitá INFM-Coherentia, Via Cintia, I-80126 Naples, Italy

Dietrich Stauffer

  • Institute for Theoretical Physics, Cologne University, D-50923 Köln, Germany

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Issue

Vol. 70, Iss. 5 — November 2004

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