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 Λℓ. This relation is used to calculate the bond percolation threshold on Λℓ. 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 Λℓ. This relation is used to calculate the bond percolation threshold on Λℓ. 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 -