Some Exact Results on Bond Percolation

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 $\Lambda$ by $\ell$ bonds connecting the same adjacent vertices, thereby yielding the lattice ${\Lambda }_{\ell }$. This relation is used to calculate the bond percolation threshold on ${\Lambda }_{\ell }$. We show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality $d\ge 2$ 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 $N\to \infty$ 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 $N\to \infty$.

33 pages latex 3 figures

References in corpus (8)

• Exact bond percolation thresholds in two dimensions
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Cited by in corpus (1)

• Exact Results for Average Cluster Numbers in Bond Percolation on Infinite-Length Lattice Strips