"strongly mixing" action on dimers?

Question

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}) $

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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 ? $$ H = \frac{1}{k^2} \log |\#\{ \text{ tilings of }\square_{k \times k}\}|$$

Answer 1

Let's start with Ian's example of $\{ 0 , 1 \}^{\mathbb{Z}}$. I'll write a point of $\{ 0,1 \}^{\mathbb{Z}}$ as a doubly infinite sequence $(x_i)$. The topology on $\{ 0,1 \}^{\mathbb{Z}}$ has a basis of open sets of the form $\Omega(L,R, (a_L, \ldots, a_R)) := \{ (x_i) : (x_L, x_{L+1}, \ldots, x_R) = (a_L, a_{L+1}, \ldots, a_R) \}$ for some integers $L \leq R$ and some fixed bit sequence $(a_L, \ldots, a_R)$. The measure of this open set is $1/2^{R-L+1}$. Intuitively, we are talking about independent random coin flips.

Saying that translation $T$ is strongly mixing means that, for any $(L,R, (a_L, \ldots, a_R))$ and $(L', R', (a_{L'}, \ldots, a_{R'}))$, the probability that $x \in \Omega(L,R, (a_L, \ldots, a_R)) \cap T^N \Omega(L',R', (a_L', \ldots, a_R'))$ approaches $2^{-(R-L+1)} 2^{-(R'-L'+1)}$ as $N \to \infty$. Sure enough, as soon as $N$ is large enough that the intervals $[L,R]$ and $[L'+N, R'+N]$ don't overlap, the bits in those intervals become independent.

Note that it is exactly the fact that translation is NOT mixing on $\mathbb{Z}$ that allows it to be mixing on $\{ 0,1 \}^{\mathbb{Z}}$. If translation were mixing on $\mathbb{Z}$, then arbitrary translates of $[L',R']$ would continue to intersect $[L',R']$, so the bit strings would be correlated on these intervals and the probabilities wouldn't become independent.

Dimers are more complicated, because the condition that the region outside two planar patches is tileable imposes constraints even when the regions don't overlap. But it is intuitively plausible that this correlation decays as the regions become further separated, and the equation you quote makes this precise.

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Your stated goal is true, and is pointed out by Kenyon immediately after Theorem 1. He expresses the frequency of motif with $k$ dimers as a $k \times k$ determinant, each entry of which is in $\mathbb{Q} + \mathbb{Q} (1/\pi)$.