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Rigidity of the interface for percolation and random-cluster models

Guy Gielis·Geoffrey Grimmett
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Abstract

We 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.

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Guy Gielis, Geoffrey Grimmett. 2001-09-17. Rigidity of the interface for percolation and random-cluster models. https://arxiv.org/abs/math/0109103

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