A note on disagreement percolation (Q2725032)

The author constructs a counterexample for the independence assumption in the disagreement percolation method for proving Gibbsian uniqueness. Therefore it turns out that the independence of the two configurations \( X_1 \) and \( X_2\) with respective distributions \( \mu_1 \) and \( \mu_2 \) cannot be dropped. In the result of \textit{J. van den Berg} [Commun. Math. Phys. 152, No.~1, 161-166 (1993; Zbl 0768.60098)] for disagreement percolation on an infinite locally finite graph \( G \) one has uniqueness of two Gibbs measures \( \mu_1 \) and \( \mu_2 \) for the same specification of a Markov random field on \( G \) taking values in a finite set \( S \). More precisely, if there is no infinite path of disagreements between \( X_1 \) and \( X_2\), i.e. NEWLINE\[NEWLINEP(G \text{contains an} \infty\text{-path of disagreements between} X_1 \text{and} X_2)=0,\tag{1}NEWLINE\]NEWLINE then \( \mu_1=\mu_2 \). One might believe that one gets uniqueness for any coupling of \( \mu_1 \) and \( \mu_2 \). The author presents an example where still (1) is valid but where there is no independence in the coupling and \( \mu_1\not=\mu_2 \). The example is the Ising model on a graph which is obtained by local modifications of the square lattice \( \mathbb{Z}^2 \). The modification the author uses is the \( (23,3)\)-decoration of \( \mathbb{Z}^2\), where for positive integers \( n \) and \( k \) the \( (n,k)\)-decoration of a graph is the graph obtained by replacing each edge of it by \( n \) parallel paths, each of length \( k \). In the last section a counterexample in the classical theory of couplings of Markov chains is presented.