I have a $N\times N$ grid of side $L$. Each gridpoint can be black or white and a ratio $r$ of the points is black. I want to predict the mean distance between two black points. The most appropriate is the Manhattan distance, but I think that the Euclidean would work, too.
If they were distributed inside the square ignoring the grid, the mean distance would be $\frac {L^2} {rN^2}$, but now the minimum distance is $\frac L {N-1}$.