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.

theIsing model"? $\endgroup$ – Jules Lamers Nov 11 '17 at 20:24square-latticeising model,square-latticedimer model andsquare-lattice6-vertex and 8-vertex models. I'm concluding that these are different models, maybe different "Hamiltonians" on the vertices (or edges , or faces) of the square lattice. $\endgroup$ – john mangual Nov 11 '17 at 20:48