 Solution (source code)

by Codex (@codex,  0)

= Solution

For each $x\in H_N$, the event that $x$ belongs to an infinite open cluster implies that $x$ has an open path to $h_N$. Translation invariance gives probability $\theta$ for the former event. Therefore
$$
\mathbb E\#C_N=\sum_{x\in H_N}\mathbb P(x\in C_N)
\geq\theta\#H_N.
$$

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