Rigidity of the interface in percolation and random-cluster models

We study conditioned random-cluster measures with edge-parameter p and cluster-weighting factor q satisfying q ≥ 1. The conditioning corresponds to mixed boundary conditions for a spin model. Interfaces may be defined in the sense of Dobrushin, and these are proved to be ‘rigid’ in the thermodynamic limit, in three dimensions and for sufficiently large values of p. This implies the existence of nontranslation-invariant (conditioned) random-cluster measures in three dimensions. The results are valid in the special case q = 1, thus indicating a property of threedimensional percolation not previously noted.