Are periodic tilings stable against defects outside of some region away from the defect?

Question

Suppose that I have a set of Wang tiles on a 2D infinite grid, and that normally the tiling pattern is periodic. Assume it has period $p$ in both vertical and horizontal direction. Then at fixed points on the lattice, I remove the tile/introduce a defect. Is there any notion of the periodic tiling recovering its structure sufficiently far away from the defects?

There appears to be a similar result for types of aperiodic tilings. Here it's is shown that a Robinson tiling is stable with respect to defects: https://www.mimuw.edu.pl/~miekisz/stablejsp.pdf

Answer 1

As I understand your question, you are asking:

Suppose we have a finite set of Wang tiles, which have a periodic tiling $T$ of the plane. Now, fix the colors on finitely many grid squares, such that the result can still be extended to a tiling of the plane. Can this extension can be chosen such that the tiling is identical to $T$ up to a translation, except for finitely many errors near the defect?

The answer to this question is "no". Consider the eight Wang tiles given by rotations of the following two tiles:

Observe that in any tiling of these, the green squares must be matched up, so we are using domino shapes with certain matching conditions. The resulting dominoes look like this, up to rotation:

where the grey circles indicate a side that can be either red or blue, whichever we choose.

After some thought, it should be clear that the only restriction this imposes on a domino tiling is that no two dominoes can be aligned edge-to-edge, with their union forming a $2\times 2$ square. Note that every arrangement of these Wang tiles uniquely determines a squarefree domino tiling of the plane, and every squarefree domino tiling yields at least one corresponding arrangement of the Wang tiles.

First, note that these tiles have a periodic arrangement:

(I've shown the domino tiling, but it should be obvious how to convert this into a periodic arrangement of the Wang tiles.)

On the other hand, suppose that we start off with a domino configuration like this:

Then it is not too hard to verify that the unique squarefree domino tiling containing this arrangement looks like:

But no translation of this latter tiling will ever line up with our periodic one! So we cannot place Wang tiles beyond those original eight in a way that reverts to our original periodic tiling. (In fact, we only need to fix three of them for this behavior to occur.)