Gibbsianness and non-Gibbsianness in divide and color models

Bálint, András

Ann. Probab., Tome 38 (2010) no. 1, p. 1609-1638 / Harvested from Project Euclid

Résumé

For parameters p ∈ [0, 1] and q > 0 such that the Fortuin–Kasteleyn (FK) random-cluster measure Φ_p,q^{ℤ^d} for ℤ^d with parameters p and q is unique, the q-divide and color [DaC(q)] model on ℤ^d is defined as follows. First, we draw a bond configuration with distribution Φ_p,q^{ℤ^d}. Then, to each (FK) cluster (i.e., to every vertex in the FK cluster), independently for different FK clusters, we assign a spin value from the set {1, 2, …, s} in such a way that spin i has probability a_i. ¶ In this paper, we prove that the resulting measure on spin configurations is a Gibbs measure for small values of p and is not a Gibbs measure for large p, except in the special case of q ∈ {2, 3, …}, a₁ = a₂ = ⋯ = a_s = 1/q, when the DaC(q) model coincides with the q-state Potts model.

Publié le : 2010-07-15
Classification: Divide and color models, Gibbs measures, non-Gibbsianness, quasilocality, random-cluster measures, 60K35, 82B20, 82B43

@article{1278593962,
     author = {B\'alint, Andr\'as},
     title = {Gibbsianness and non-Gibbsianness in divide and color models},
     journal = {Ann. Probab.},
     volume = {38},
     number = {1},
     year = {2010},
     pages = { 1609-1638},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1278593962}
}

Bálint, András. Gibbsianness and non-Gibbsianness in divide and color models. Ann. Probab., Tome 38 (2010) no. 1, pp.  1609-1638. http://gdmltest.u-ga.fr/item/1278593962/