# Uniqueness of Gibbs Measure on Ising model

If I understood this correctly, the Gibbs Specification for the Ising model on $ℤ^d$ dos not have a unique Gibbs Measure for β above the critical level. But what about the Ising model on a finite lattice $V⊂ℤ^d$?

And if it is not unique for a finite lattice, how is that possible as I can construct a Gibbs Sampler – a markov chain, that has the measure that fulfills the specification as its stationary distribution – that is clearly irreducible (we can go from anywhere to the $\{-1\}^V$ state and get everywhere from there) and thus must have an unique stationary distribution?

• Too much to parse to me. – Włodzimierz Holsztyński Sep 23 '13 at 4:20
• @WlodzimierzHolsztynski ? – Chronial Sep 23 '13 at 5:56
• Your second (last) question is difficult to follow to me. (BTW, it'd be nice to see the relevant definitions. I'd like such definition-oriented spirit in general on MO). – Włodzimierz Holsztyński Sep 23 '13 at 6:29
• Isn't the Gibbs measure in a finite lattice (at the given temperature) unique by the very definition? – Włodzimierz Holsztyński Sep 23 '13 at 16:53
• You @Chronial remind me of a Turing secretary. – Włodzimierz Holsztyński Sep 23 '13 at 17:09