Self-dual planar hypergraphs and exact bond percolation thresholds

TL;DR

This paper explores self-dual planar hypergraphs, providing a mathematical framework for exactly determining bond percolation thresholds and revealing the existence of infinitely many thresholds with corresponding lattices.

Contribution

It introduces a new class of hypergraphs and clarifies misconceptions, expanding understanding of percolation thresholds in lattice models.

Findings

Existence of infinitely many real numbers with corresponding lattices

Mathematical basis for exact bond percolation thresholds

Clarification of misconceptions in prior research

Abstract

Recent research on percolation has led to the construction of an infinite class of lattices for which the percolation thresholds can be determined exactly. We discuss the mathematical basis for the solutions of bond percolation models, and, in particular, attempt to address some misconceptions that might arise from the seminal articles by Ziff and Scullard. As one consequence, we show that there exist infinitely many real numbers $a \in [0,1]$ for which there are infinitely many lattices that have bond percolation threshold equal to $a$.