## 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 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→∞.

Original language | English |
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Pages (from-to) | 676-700 |

Number of pages | 25 |

Journal | Journal of Statistical Physics |

Volume | 149 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2012 Nov 1 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics