Regions in infinite random grids

Here is a puzzle I thought of:

What is the average area of a contiguous region in a infinite, random binary grid?

Unfortunately, I can’t solve it. The 1d case has been solved by others, and is a good warmup. The trick is to write an infinite sum, and when you do, you should get 2. In 2d it’s much harder to find such a sum, so I tried it numerically.

The idea is to sample many random n-by-n grids for increasing values of n, and count the average size of a region. For example, here is a binary grid of side-length n = 5:

This example has 6 regions, so the average area in this sample is 25/6. Perhaps it’s clearer if I color them:

For several values of n, I’m going to generate many such grids and plot the results. The idea is that the average should converge to the correct answer in the limit n→∞.

I’ve also plotted statistical errors. From this plot, it appears that the answer is about 7, or perhaps it diverges logarithmically? Let me know if you have any ideas.

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