\documentclass[12pt]{article}
\usepackage{amsmath,mathrsfs,bbm}
\usepackage{amssymb}
\textwidth=4.825in
\overfullrule=0pt
\thispagestyle{empty}
\begin{document}
\noindent
%
%
{\bf John C. Wierman and Robert M. Ziff}
%
%
\medskip
\noindent
%
%
{\bf Self-dual Planar Hypergraphs and Exact Bond Percolation Thresholds}
%
%
\vskip 5mm
\noindent
%
%
%
%
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~$a$.
\end{document}