What is the average size of all continuous unicolored areas in a randomly colored grid?

Question

It is somewhat similar to the percolation theory, but I can't find any reference for what happens with multiple colors involved.

Let there be an infinite grid and $n$ colors. Each tile in the grid is assigned a random color (each color with probability $1/n$). What is the average size of all unicolored areas? (We always consider the biggest possible unicolored area).

I am most interested in a solution for $n = 3$.

I am worried that there will not be any nice closed form solution, but I do hope that it will be something beautiful involving $e$.

Answer 1

It really is a case of (site) percolation. Let $A$ be the largest connected unicoloured region containing a particular tile. This is equivalent to the percolation cluster containing an open site, where each site is open with probability $1/n$. You just identify an open site with a tile of the same colour as your particular tile.

Answer 2

So I made some simulations and dug a bit into this. Let's see whether someone stumbles upon this question.

The answer for n=3 is ~2.695. It is very close to $e$, but not quite. The average size only rarely exceeds $e$ with simulation on 1000x1000 grid. It indeed is very consistently around 2.695. I don't have a closed form solution, and I am now convinced that there is not one.

Average area for n=2 is ~7.55 and for n=4 is ~1.94

Read more in my blogpost.