TY - JOUR

T1 - Some Exact Results on Bond Percolation

AU - Chang, Shu Chiuan

AU - Shrock, Robert

N1 - Funding Information:
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.).

PY - 2012/11

Y1 - 2012/11

N2 - 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 ℓ bonds connecting the same adjacent vertices, thereby yielding the lattice Λ ℓ. This relation is used to calculate the bond percolation threshold on Λ ℓ. 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→∞.

AB - 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 ℓ bonds connecting the same adjacent vertices, thereby yielding the lattice Λ ℓ. This relation is used to calculate the bond percolation threshold on Λ ℓ. 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→∞.

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U2 - 10.1007/s10955-012-0616-5

DO - 10.1007/s10955-012-0616-5

M3 - Article

AN - SCOPUS:84870360802

SN - 0022-4715

VL - 149

SP - 676

EP - 700

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 4

ER -