Percolation and the Potts Model

The percolation process provides a simple picture of a critical point transition and has been of increasing recent theoretical interest. We refer to several review articles (1-a~ for a general survey of the subject. An important development first established by Kasteleyn and Fortuin (4,5~ is the connection between bond percolation and a lattice statistical model. This consideration leads to a formulation of the percolation problem which is extremely useful, for many of the techniques readily available in statistical mechanics can now be applied to percolation (see, e.g., Ref. 6). However, much of this otherwise elegant result appears to be masked under the formality of the graph-theoretic approach of Kasteleyn and Fortuin, and we feel it worthwhile to have an alternate derivation to elucidate the situation. We present an approach to the Kasteleyn-Fortuin formulation of bond percolation which we believe to be simpler and more direct; it also permits a straightforward extension of the Griffiths inequality to the percolation problem. In Section 2 we review the bond percolation problem for the purpose of establishing the notation. The Potts model is introduced in Section 3, and we show that the quantities of interest arising in the percolation problem, including the critical exponents,