Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 2 / a1

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- Solution a1

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

θ

for the former event. Therefore

E # C_{N} = \sum_{x \in H_{N}} P (x \in C_{N}) \geq θ # H_{N} .

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