Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 214 / 2 / a

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- Solution a

Solution

Write

N_{m} = ∣ B (m) ∣

and

θ = θ (p) > 0

. By translation invariance and linearity of expectation,

E_{p} R (m) = \sum_{x \in B (m)} P_{p} (x belongs to the infinite cluster) = θ N_{m} .

Let

r = P_{p} (R (m) \geq θ N_{m} /2)

. Since

0 \leq R (m) \leq N_{m}

, splitting the expectation at the threshold gives

θ N_{m} \leq \frac{θ N _{m}}{2} (1 - r) + N_{m} r .

Rearranging produces the slightly stronger estimate

P_{p} (R (m) \geq \frac{θ ( p ) ∣ B ( m ) ∣}{2}) \geq \frac{θ ( p )}{2 - θ ( p )} \geq \frac{θ ( p )}{2} .

Only the mean and boundedness of the random variable are used; independence of the vertex-membership events is unnecessary.

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