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I'd like to ask questions about a "random domino tiling of the plane". However, it's not quite obvious how to go about precisely specifying what this means.

My first instinct was to do something like "the center of a random tiling of a large square". More formally, consider the following property for a random distribution on grid-aligned domino tilings in the plane:

For each $r>0$, the distribution of domino positions within distance $r$ of the origin is the limit of the distributions on that region among randomly-selected tilings of $2N\times 2N$ origin-centered squares as $N$ goes to infinity.

Clearly, if a distribution satisfying this property exists, it is unique, and seems like a reasonable definition to use. However, while intuitive, it is not clear to me how to show that the centers of random tilings of large squares converge in the necessary sense.

The most pressing question about this distribution is whether it actually exists, which I strongly suspect is the case. Conditional on that being true, I have several followup questions:

  • Is the same distribution obtained if we replace "square" with "torus" or "aztec diamond"? Random tilings of the latter are substantially easier to describe and generate (see the Arctic Circle theorem).

  • Is the distribution translation-invariant? Again, I strongly suspect this is the case, but I don't know how I'd go about proving it. If so, how far into a large square do we need to go to see this distribution? E.g., is it the case that a patch of a random $2N\times 2N$ tiling at distance $\log(N)$ from the border asymptotically looks like a patch at the center?

  • What exact probabilities of different configurations does it have? For instance, what are the odds that the origin is not a vertex of any domino? Concretely, at the center of a $2k\times 2k$ grid we have probabilities of $1,\frac19,\frac{361}{841},\frac{139129}{811801},\ldots$, which is approximately $1,0.1111,0.4293,0.1714,\ldots$.

  • Is there a reasonably efficient algorithm to sample from finite portions of this distribution? Ideally in a constructive manner, i.e. an algorithm which when initialized with a random seed spits out progressively more and more dominoes around the origin that extend to a tiling of the plane.

Possibly relevant is the fact that a random tiling of a $2N\times 2N$ square is given in the limit by starting with any tiling and randomly flipping any two dominoes joined in a $2\times 2$ square (see e.g. Laslier and Toninelli 2012).

Update: I have cross-posted this question to MathOverflow here.

RavenclawPrefect
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    Kasteleyn's article already speaks about infinite configurations, so this certainly should exist in literature. It also discusses toric configurations, and I think it should be possible to conclude from there that infinite toric domino tilings are different. For Aztec diamond, this should certainly be very different. Maybe, it is better to ask on math.MO. – zhoraster Jan 28 '21 at 18:45
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    @zhoraster: Thanks for the link! Looking at Table 2 in that paper, they compute identical molecular freedom of the square and toroidal cases in the limit, which seems suggestive of convergence? (Though I also don't see why these statistics differing would imply a different final distribution.) Can you elaborate on why you expect the Aztec diamond to behave differently? And yes, I plan to ask on MathOverflow once the bounty expires - I was hesitant to post without first confirming that there was not some trivial resolution to the problem. – RavenclawPrefect Jan 28 '21 at 19:04
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    I didn't say that Kasteleyn proves existence of the limit, but that he already speaks about that, so I guess in 70 years following his article someone should have written about that. I am not very sure that in the toric infinite case is different, just think that the results might already allow concluding that. Concerning the Aztec diamond, it is also just intuition, since it's too different. I am no specialist here at all; I might try to go deeper into topic and try to answer your questions, but it's better to ask specialists :) – zhoraster Jan 28 '21 at 19:16
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    What about trying to do the limit only on the number of dominos? In this way you don't have any "boundary problem" (it could be the case that a given tiling on a small region does not extend to a square, because of the boundary conditions). Also, if you take a finite region $R$ that can be covered with at most $n(R) $ pieces, the distribution for $m > n(R) $ should be stationary. Am I missing something stupid? – Andrea Marino Jan 28 '21 at 23:45
  • @AndreaMarino: Every finite patch of dominoes that extends to a tiling of the plane does extend to a tiling of some square, but I agree this is non-obvious; the only proof I know is rather nontrivial. It's not clear to me how one would take a limit for a fixed number of dominoes, because there are infinitely many arrangements of $k$ dominoes for any fixed $k$. As for the distribution beyond a given region, I'm not sure I follow your meaning? – RavenclawPrefect Jan 29 '21 at 04:57
  • As far as I understood, you want to define a distribution on tilings that covers a given finite region $R$. Such tilings (I assume) have integer-coordinates-dominoes, so that there is only a finite number of arrangements of at most $k$ dominoes that covers $R$, and such that every piece intersects $R$. You have to put $k$ variable just because the region could be irregular, so that can be covered by a different number of dominoes; but for big $k$ using at most k pieces or at most (say) 100 pieces is the same. – Andrea Marino Jan 29 '21 at 08:58
  • Another problem I see with the square approach is the following: it could be that squares prefer some kind of configurations in its center; this is not about tilings of the region but about square tastes. If you then substitute with an Aztec diamond (say) it could prefer different tilings. For example, take a 4x4 square and look at the tilings it induces in the central 2x2 square: if I am not wrong there are 4 possible outcomes, weighted 2,2,1,1 , while I would expect the four possibilities to be equally possible. – Andrea Marino Jan 29 '21 at 09:07
  • @AndreaMarino: I count $34$ possible ways to cover a $2\times 2$ square with dominoes, of which $32$ have exactly one extension to a $4\times 4$ tiling and $2$ have two such extensions (the ones with two extensions are those which cover the $2\times 2$ exactly, with no excess). This matches the $36$ possible tilings of a $4\times 4$ square with dominoes. But it is not clear to me why these four possibilities should be equally likely; if they were, it would necessarily be the case that the possible induced coverings of a $4\times 4$ square were unequally distributed. – RavenclawPrefect Jan 29 '21 at 17:43
  • i considered two tilings equivalent if they induce the same partition on the interior of the 2x2 square; we had different conventions. As for your second objection, I don't see why a distribution on the 2x2 square should be influenced by the distribution on a bigger square: this is something you decided. But for example what if we take an irregular region? what the distribution of this should be linked? – Andrea Marino Jan 29 '21 at 18:01

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