2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62203We study conditioned random-cluster measures with edge-parameter p and cluster-weighting factor q satisfying q \ge 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 non-translation-invariant (conditioned) random-cluster measures in three dimensions. The results are valid in the special case q=1, thus indicating a property of three-dimensional percolation not previously noted.33 pagesProbabilityMathematical Physics60K35, 82B20, 82B43text