Rigidity of the Interface in Percolation and Random-Cluster Models |
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Authors: | Guy Gielis Geoffrey Grimmett |
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Institution: | (1) King's College Research Centre, Cambridge, CB2 1ST;(2) Statistical Laboratory, Centre for Mathematical Sciences, Cambridge, CB3 0WB, United Kingdom |
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Abstract: | 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 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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Keywords: | Random-cluster model percolation Ising model Potts model interface Dobrushin boundary condition |
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