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).

This was previously posted on Math StackExchange here, without much progress. In the comments, a paper of P. W. Kasteleyn was linked which may be relevant.