Exact bond percolation thresholds in two dimensions

Abstract

Recent work in percolation has led to exact solutions for the site and bond critical thresholds of many new lattices. Here we show how these results can be extended to other classes of graphs, significantly increasing the number and variety of solved problems. Any graph that can be decomposed into a certain arrangement of triangles, which we call self-dual, gives a class of lattices whose percolation thresholds can be found exactly by a recently introduced triangle-triangle transformation. We use this method to generalize Wierman's solution of the bow-tie lattice to yield several new solutions. We also give another example of a self-dual arrangement of triangles that leads to a further class of solvable problems. There are certainly many more such classes.

Accepted for publication in J. Phys A

References in corpus (3)

• Clique percolation in random networks
• Tunneling and Non-Universality in Continuum Percolation Systems
• Predictions of bond percolation thresholds for the kagomé and Archimedean $(3,12^2)$ lattices

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• Percolation thresholds on 2D Voronoi networks and Delaunay triangulations
• Critical surfaces for general bond percolation problems
• Universal condition for critical percolation thresholds of kagome-like lattices
• Percolation in Networks with Voids and Bottlenecks