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Journal of Statistical Physics

, Volume 149, Issue 4, pp 676–700 | Cite as

Some Exact Results on Bond Percolation

  • Shu-Chiuan Chang
  • Robert Shrock
Article

Abstract

We present some exact results on bond percolation. We derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice Λ by bonds connecting the same adjacent vertices, thereby yielding the lattice Λ . This relation is used to calculate the bond percolation threshold on Λ . We show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality d≥2 but changes it on a one-dimensional lattice and quasi-one-dimensional infinite-length strips. We also present analytic expressions for the average cluster number per vertex and correlation length for the bond percolation problem on the N→∞ limits of several families of N-vertex graphs. Finally, we explore the effect of bond vacancies on families of graphs with the property of bounded diameter as N→∞.

Keywords

Bond percolation Potts model 

Notes

Acknowledgements

This research was partially supported by the Taiwan National Science Council (NSC) grant NSC-100-2112-M-006-003-MY3 (S.-C.C.) and by the U.S. National Science Foundation grant NSF-PHY-09-69739 (R.S.).

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Physics DepartmentNational Cheng Kung UniversityTainanTaiwan
  2. 2.C.N. Yang Institute for Theoretical PhysicsStony Brook UniversityStony BrookUSA

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