Some Exact Results on Bond Percolation

TL;DR

This paper provides exact analytical results on bond percolation, including thresholds, universality, and effects of bond inflation and vacancies on various lattice and graph structures.

Contribution

It introduces a relation for bond inflation effects, computes percolation thresholds, and analyzes the impact of bond vacancies on large graph families.

Findings

Bond inflation preserves universality class in dimensions ≥ 2

Exact bond percolation thresholds derived for inflated lattices

Bond vacancies significantly affect percolation properties in bounded diameter graphs

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 $N \to \infty$.