# Translationally invariant Gibbs state

In Velenik's Statistical Mechanics of Lattice Systems, Exercise 6.22 claims that if $$\pi =\{ \pi_\Lambda:\Lambda \Subset\mathbb{Z}^d\}$$ is a translationally invariant specification with nonempty Gibbs state $$\mathscr{G}(\pi)$$ (probability measures compatible with $$\pi_\Lambda$$), then the set of translationally invariant Gibbs state is nonempty.

Velenik provides a hint in which we take $$\mu\in \mathscr{G}(\pi)$$ and use $$\mu_n = \frac{1}{|B(n)|} \sum_{j\in B(n)} \theta_j \mu$$

I would assume that we are to take the limit (passing under subsequence by Banach-Alaoglu) and obtain a vague convergence probability measure, but it seems that this proof requires that $$\mathscr{G}(\pi)$$ be closed under the vague topology, which may not necessarily be true. Is there another way to prove the claim? Or is the statement missing a requirement?

## 1 Answer

You are completely correct. We should have added the assumption that the specification $$\pi$$ is quasilocal. (For instance, using the setting described right before the exercise; I guess that's what we had in mind when stating the exercise, but I don't really remember. Of course, in that case, one can remove the assumption that $$\mathcal{G}(\pi)\neq\emptyset$$ by Theorem 6.26.)

With the quasilocality assumption, $$\mathcal{G}(\pi)$$ is closed by Lemma 6.27 and the exercise is proved as described in the book (solutions are given in Appendix C).

I have updated the errata and corrected the "preprint" version of the book. (Thanks for pointing this out!)

(By the way, if you are interested in the corresponding result in a much more general framework than we cover in our book, you should have a look at Corollary 5.16 in Georgii's book.)

• Thank you for the recommendation! I have seen Georgii's book before, but it is a little too general for me at the moment. Sep 30, 2020 at 21:38
• @AndrewYuan The fact that Georgii's book is too abstract to learn the theory from it is one of the reasons we decided to write our book: there was a need for an introductory book on this subject. ;) Oct 1, 2020 at 6:03