Percolation and ferromagnetism on \(\mathbb Z^ 2\): the \(q\)-state Potts cases
The relation between the existence of percolation in a limiting Gibbs state under a constant boundary condition (``green'') and the phase transition is investigated to get results analogous to those obtained by \textit{A. Coniglio, C. R. Nappi, F. Peruggi} and \textit{L. Russo} [Commun. Math. Phys. 51, 315-323 (1976) and J. Phys. A 10, 205 (1977)] for the Ising model. It is shown that if there is no infinite cluster of ``green sites'' in the limiting ``green'' state, then the Gibbs state is unique. If the limiting ``green'' state has an infinite cluster, then we are below or at the transition temperature. If we are at the transition temperature and there is the order/disorder coexistence, then the ``green'' state contains an infinite cluster of ``green sites'' whereas the limiting state under the free boundary condition contains no infinite cluster. The problem is reduced to methods used by \textit{A. Gandolfi, M. Keane} and \textit{L. Russo} [Ann. Probab. 16, No. 3, 1147-1157 (1988; Zbl 0658.60133)] and by \textit{M. Aizenman, J. T. Chayes, L. Chayes} and \textit{C. M. Newman} [J. Stat. Phys. 50, 1-40 (1988)] and to the proof of FKG inequality for some auxiliary measures.
scientific article; zbMATH DE number 1134797