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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?

(A followup question would be how many connected componants we can expect provided this set up on an $N$ by $M$ grid...)

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?

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?

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 the 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?

(A followup question would be how many connected componants we can expect provided this set up on an $N$ by $M$ grid...)

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?

(A followup question would be how many connected componants we can expect provided this set up on an $N$ by $M$ grid...)

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 the 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?

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 the 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?

(A followup question would be how many connected componants we can expect provided this set up on an $N$ by $M$ grid...)