Boundary Conditions for Translation-Invariant Gibbs Measures of the Potts Model on Cayley Trees

We consider translation-invariant splitting Gibbs measures (TISGMs) for the q -state Potts model on a Cayley tree of order two. Recently a full description of the TISGMs was obtained, and it was shown in particular that at sufficiently low temperatures their number is 2^q-1 . In this paper for each TISGM μ we explicitly give the set of boundary conditions such that limiting Gibbs measures with respect to these boundary conditions coincide with μ .