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"?