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Supose each cell of a big (or infinite) grid is colored at random by one of $k$ colors. Then the connected monochromatic components (here components are not supposed to contain "wasp waists", i.e., such a grid point is considered to separate two components) define a tiling of the plane. What is the average size of each such component?

I am sure this has been examined before, but I can't find anything, and I don't have an idea how to tackle that "globally", without tedious local case-by-case considerations. The question can obviously be generalized from square cells to any kind of periodic (or even aperiodic, like e.g. Penrose-style) tiling of the plane, but also to grids / tilings in higher dimensions, which looks like hitting a whole beehive.

So I would like to ask the question first for the tilings of the plane by squares, triangles, hexagons or Penrose kites & darts.
Is it possible that the average component size for given $k$ and any tiling depends only on the average number of tiles meeting at each "corner"?

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I think it's very unlikely that this depends only on the average degree as you suggest.

Equivalently, you're asking for the average size of a component in vertex- (or site-) percolation on the dual of your tiling at parameter $1/k$. Average degree in the prime is a slightly unnatural property in the dual, but if instead I look at the average degree in the dual we can see from Wikipedia's table of percolation thresholds that there are multiple 3-regular lattices with differing percolation thresholds. In particular, there are values of the parameter for which some lattices will have infinite components and other lattices won't. No infinite component is not the same as finite average component size, degree in the dual isn't the specific parameter you were interested in and the thresholds are large enough that $1/k$ is generally below all of them, but I don't expect that your original question is different enough that your suggested criterion for average component size can work.

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