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Some Exact Results on Bond Percolation
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 by bonds connecting the same adjacent
vertices, thereby yielding the lattice . This relation is used to
calculate the bond percolation threshold on . We show that this
bond inflation leaves the universality class of the percolation transition
invariant on a lattice of dimensionality 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
limits of several families of -vertex graphs. Finally, we explore the effect
of bond vacancies on families of graphs with the property of bounded diameter
as .Comment: 33 pages latex 3 figure
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