TITLE: What is the link between the Domino Tilings and the Ising Model?
QUESTION [7 upvotes]: Is there a link between the theory of Domino Tilings and the Ising Model?   In the global qualitative sense that physicists use, the answer is "yes".  The connections could go like this:

The dimer model is the limit of the ising model (Onsager?)
The theory of domino tiling is dual to the theory of dimer tilings. (Kenyon?)

However, when I put the two connections together, I wasn't able to say this Ising model configuration maps to this Domino Tiling.  And I suspect, they are slightly different objects.  First here's the section from Baxter's book on Exactly Solved Models giving us hope:

He tells us, the Ising Model "freezes" into a Dimer problem, which is dual to a domino tiling problem.
From Fisher's original 1966 paper we find, the square Ising model maps to dimers model on "decorated" square lattice:

The theories of exactly solved statistical mechanics and of integrable systems are not necessarily related.  It's just that if I read one, then I inadvertently start reading the other.  Here's a discussion of Korepin and Izergin.

So now, there is also a link between 6-vertex model and domino tilings.  Does that also have fine print? 

For all of the above questions, I cited papers from the 1960's and 1980's.  I wonder what the modern perspective of integrable theories are.
I seem to have asked a question like this already.  And there will be more clarifications yet to come.

map from 6-vertex model to domino tiling

REPLY [2 votes]: Since others have already commented on the link with domino tilings, let me point out a relation between the eight-vertex model and an Ising-type model on the square lattice, cf the comments.
These Ising-type models are a bit unusual: they feature both two- and four-point interactions. [The latter are like the interactions of an interaction-round-a-face (IRF) model on the dual lattice.] The weights can be made to match, and the relation is 2-to-1 at the level of configurations, so the partition functions match up to a factor of two. See e.g. eight-vertex model at Wikipedia and the references there.
(As you know I explained the relation between the six-vertex model and domino tilings of the Aztec diamond, cf the last figure in the OP, in my answer to your question that you mention.)