In Local Statistics of Lattice Dimers we study a nice familiar object, domino tilings in the plane extending out to infinity.

His paper is going to discuss the frequency of various "motifs" in tiligns of infinite dimer regions and I am hoping to verify all such frequences lie in $\mathbb{Q}[\tfrac{1}{\pi}]$.

However, I am thrown off by the following phrase:

It is known that the translation-action of $\mathbb{Z}^2$ on dominos or lozenges is topologically mixing

I didn't even know where was a translation action on domino tilings. Kenyon proceeds to quantify this "mixing" as

$$ \mu(U_{T_1}\cap U_{v+T_2}) = \mu(U_{T_1})\,\mu(U_{v+T_2}) + O(\tfrac{1}{|v^2|}) $$

Apparently the translation $T_v: \mathbb{Z}^2 \to \mathbb{Z}^2$ induce a translation on the set of infinite dimer tilings on the plane $T_v: X \to X$.

This mixing result says any two motifs $T_1, T_2$ are relatively independent of each other. Then a tiling is expected to have a motif $T_1$ in one area with probability $\mu(U_{T_1})$ and $T_2$ with probability $\mu(U_{T_2})$. The odds of having both is close to $\mu(U_{T_1})\mu(U_{T_2}) $

Intuitively, I would have just embedded both regions $T_1$ and $T_2$ in a tilings of the square and computed the frequences of both motifs $T_1$ and $v + T_2$ occurring in random tilings as the size of the square got large.

However, the more they try to get precise, the more confused I get.

- Translation is not mixing in the plane, and yet it is mixing on the space of domino tilings. How does that make sense?
- What was the important of identifying the
**unique**measure $\mu$ of maximal entropy? and showing $\mu$ was the same as that of uniform random tiling for large squares with entropy ? $$ H = \frac{1}{k^2} \log |\#\{ \text{ tilings of }\square_{k \times k}\}|$$

open sets of configurations(which is what occurs in the definition of topological mixing) with the effect of that action on anindividualconfiguration (an unilluminating isometry on $\mathbb{Z}^2$: but the action is typicallynotan isometry on the configuration space). The meaning of "mixing" in this context is thatopen setsof configurations will tend to be brought, by the action, into intersection with one another. $\endgroup$ – Ian Morris Jun 2 '14 at 12:27