Consider independent bond percolation on $\mathbb{Z}^2$, with $p>p_c$ so that the process is supercritical. For any site $x$ let $Y_x$ be the indicator of $x$ belonging to the infinite open cluster. (This can be interpreted as the probability of being infected if some site in the giant component is infected.) Let $N_d$ be the set of all points within distance $d$ of the origin in $\mathbb{Z}^2$. What is known about the joint distribution of $(Y_x)_{x\in N_d}$, rigorously or heuristically?
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