I have an $N$ by $M$ grid (a Go board for example), where for every square in the grid, I place a white stone with probability $p$ and a black stone with probability $(1-p)$. We call two white stones (or two black stones) 'connected' if there exists a path between them, consisting of stones of their same color, that hops along von Neumann neighborhood of each cell along the way (North, South, East, West nearest-neighbors). Diagonal hops are prohibited. 


What is the probability that all white stones in the $N$ by $M$ grid are 'connected' in this fashion? If all white stones are 'connected', what is the probability that each white stone has two or more white neighbors? 




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Update - The percolation threshold, $p\_c$, of a graph or lattice (hat tip to Igor Rivin) is the minimum connectivity before which one begins to see connected components spanning from one side of a graph or a lattice to the other. Is there some similar threshold for which one expects a single connected component?



Probability that a randomly filled Go board has a set of white stones connected through their von Neumann neighborhoods
The magic word is "percolation". This is a huge subject, but Grimmett's book of the same title should be a very good start.


