Rigidity of the interface for percolation and random-cluster models

2001/09/17 by Guy Gielis, Geoffrey Grimmett, Gielis, Guy +1
Mathematics · Physics and Astronomy · #60K35 #82B20 #82B43 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR #msc:60K35 #msc:82B20 #msc:82B43

paper · pdf · doi:10.48550/arxiv.math/0109103

33 pages

arxiv created 2001/09/17 · openalex publication_date 2001/09/17 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Strict inequality for critical values of Potts models and random-cluster processes

Related

• Pinning of interfaces in a random elastic medium and logarithmic lattice embeddings in percolation
• Explicit isoperimetric constants and phase transitions in the random-cluster model
• Percolation results for the Continuum Random Cluster Model
• Criticality, universality, and isoradiality
• Phase separation in random cluster models I: uniform upper bounds on local deviation