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Suppose on an infinite two dimensional Go board the tengen is occupied by a black stone, and every other grid point is occupied by a black stone, or a white stone, or nothing, with probability 1/3 respectively. What is the probability for the group belonging to the central black stone to live on the board?

More formally, suppose $f:\mathbb Z^2\to\{1,0,-1\}$ is a random function such that $f(0,0)=1$ and for all $(x,y)\neq(0,0)$ and all $a\in\{1,0,-1\}$, $\mathbb P(f(x,y)=a)=1/3$. Let $C$ be the connected component (adjacency is defined as 4 immediate neighbors, a la Von Neumann) of $f^{-1}(1)$ containing the point $(0,0)$. What is $\mathbb P(\forall (x,y)\in C', f(x,y)=-1)$, where $C'=\{(x,y)\in\mathbb Z^2\backslash C: (x,y)$ is adjacent to some point in $C\}$?

Related, but not identical: Probability that a randomly filled Go board has a set of white stones connected through their von Neumann neighborhoods

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  • $\begingroup$ See also my answer to this identical question on MSE: math.stackexchange.com/a/2803843/13524 The best way to place bounds on the result is probably to take a certain sum over all polyominoes of less than a chosen size. $\endgroup$
    – Sophie Swett
    Commented Jun 2, 2018 at 1:33
  • $\begingroup$ @TannerSwett The coincidence is due to the fact that this question was asked on zhihu (= Chinese Quora) recently. I have read (what is essentially) your answer there, but I'm still wondering what, if anything, can percolation theory say about it. $\endgroup$
    – Fan Zheng
    Commented Jun 2, 2018 at 21:08

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The probability that it is dead, i.e. is surrounded by white stones, is the sum, over all finite connected sets $C$ containing $(0,0)$, of $3^{-(|C|+|C'|)}$. I don't expect this to have a closed form. Simulation seems to indicate it is about $0.0152$. The contribution of $C = \{(0,0)\}$ to this is $3^{-4} \approx 0.0123$.

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  • $\begingroup$ I guess you mean $3^{-(|C|+|C'|-1)}$, because the black stone at (0,0) is given. $\endgroup$
    – Fan Zheng
    Commented Jun 2, 2018 at 19:48

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