Nuclear Physics B235 [FSll] (1984) 19-23 © North-Holland Pubhshing Company THE CORRECT EXTENSION OF THE FORTUIN-KASTELEYN RESULT TO PLAQUE'I~E PERCOLATION

The work of Fortuin and Kasteleyn [1] demonstrated that bond percolation may be viewed as the s ~ 1 limit of the s-state Potts model. Recent interest in theories of random surfaces, particularly in the context of gauge theories [2], has led to the question of whether there is a natural generalization of the Fortuin-Kasteleyn result to plaquette percolation. It has been asserted that the proper extension is to view plaquette percolation as the s -~ 1 limit of the gauge Potts model with s-state bond variables [3]. However , Aizenman and Fr6hlich [4] have shown that the analysis in ref. [3] is incorrect, since the constraints dictated by the gauge Potts model lead to non-trivial s dependence in the weights associated with certain surfaces. The purpose of this note is to introduce a different generalization of the Potts model for which plaquette percolation is the s ~ 1 limit. Moreover , in analogy with the results of Fortuin and Kasteleyn, the s ~ 0 limit of this model is tree percolation of plaquettes. The important point is that our model has no manifest gauge invariance; it is the natural extension of the Potts model, for which there is no difficulty in regarding s as a continuous parameter . In sect. 2, we briefly review plaquette percolation and define two order parameters which characterize distinct transitions. A model for t ree-plaquette percolation is also defined in this section. We introduce our extension of the Potts model in sect. 3 and prove that the s ~ 1, 0 limits yield plaquette percolation and tree percolation of plaquettes, respectively. The generalization of the model to r-cells in higher