Self-dual Planar Hypergraphs and Exact Bond Percolation Thresholds

Abstract

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. We investigate the mathematical conditions for the solution of bond percolation models, and identify an infinite class of lattice graphs for which exact bond percolation thresholds may be rigorously determined as the solution of a polynomial equation. This class is naturally described in terms of hypergraphs, leading to definitions of planar hypergraphs and self-dual planar hypergraphs. We show that there exist infinitely many self-dual planar 3-uniform hypergraphs, and, as a consequence, that there exist infinitely many real numbers a ∈ [0, 1] for which there are infinitely many lattices that have bond percolation threshold equal to a.

A Disproof of Tsallis' Bond Percolation Threshold Conjecture for the Kagome Lattice

“…Since then, a class of new exact solutions has been established. (See [2] and [38].) However, as pointed out by Ziff [44], the kagome lattice is not in this class, so an exact solution does not exist using current methods.…”

“…Early 1980's results [12,13,29,30] determined the exact bond percolation thresholds of the square, triangular, hexagonal and bow-tie lattices and the site percolation thresholds of the triangular and kagome lattices, using graph duality properties and the star-triangle transformation. No further progress on this problem was made until 2006, when a generalized star-triangle transformation and the concept of self-dual planar hypergraphs led to a method for solving for bond percolation thresholds for an infinite class of two-dimensional lattices [25,38,44,46]. However, this method does not produce solutions for several common models, the most well-studied being the kagome lattice bond model and the square and hexagonal lattice site models.…”

Aperiodic Non-Isomorphic Lattices with Equivalent Percolation and Random-Cluster Models

“…Given a generator G, let p 0 (G) denote the value of p such that these probabilities are equal. As a consequence of a generalised star-triangle transformation described in Wierman and Ziff [19], p 0 (G) determines the exact bond percolation threshold of the lattice constructed from a generator G as in the following paragraphs.…”

“…Research of Scullard [10], Ziff [20], Scullard and Ziff [11], and Ziff and Scullard [21] used a generalised star-triangle transformation and a concept of triangle-duality to predict exact solutions for a collection of periodic lattice graphs in two dimensions. Wierman and Ziff [19] organised this approach to describe a construction of an infinite class of planar graphs for which the bond percolation threshold can be mathematically rigorously determined. The graphs are constructed by placing isomorphic copies of a finite connected planar graph with three terminals, called a generator, in a connected self-dual planar periodic 3-uniform hypergraph structure.…”

“…A consequence of the construction is that for each of an infinite set of real numbers a ∈ (0, 1) there are a countable infinite family of periodic lattices which have bond percolation threshold equal to a. Sedlock and Wierman [12] generalised an argument of Wierman [18] using the substitution method (See [2,Ch.6]) to show that the critical exponents are equal for any dual pair of lattices constructed by the method. Placing the same generator and its dual generator in the countably infinite collection of self-dual hypergraphs identified by Wierman and Ziff [19] creates a countably infinite set of lattices which have the same critical exponents. This observation provided supporting evidence for the universality hypothesis that the critical exponents agree for all two-dimensional percolation models.…”