Self-dual Planar Hypergraphs and Exact Bond Percolation Thresholds

John C. Wierman, Robert M. Ziff


A generalized star-triangle transformation and a concept of triangle-duality have been introduced recently in the physics literature to predict exact percolation threshold values of several lattices. Conditions for the solution of bond percolation models are investigated, and an infinite class of lattice graphs for which exact bond percolation thresholds may be rigorously determined as the solution of a polynomial equation are identified. This class is naturally described in terms of hypergraphs, leading to definitions of planar hypergraphs and self-dual planar hypergraphs. There exist infinitely many self-dual planar 3-uniform hypergraphs, and, as a consequence, there exist infinitely many real numbers $a \in [0,1]$ for which there are infinitely many lattices that have bond percolation threshold equal to $a$.

Full Text: PDF