Solution

Expanding the square and using the independence of the percolation edges gives

E X_{r}^{2} = \sum_{γ, γ^{'}} μ (γ) μ (γ^{'}) \frac{P ( γ , γ ^{'} open )}{P ( γ open ) P ( γ ^{'} open )} .

The ratio equals

p^{- N_{r} (γ, γ^{'})}

, where

N_{r}

is the number of common edges in the two truncated paths. It is at most

p^{- N}

for

N = ∣ ξ \cap ξ^{'} ∣

, under the convention in the hypothesis; in particular

N \geq 1

because both paths contain

o

. For every integer

N \geq 1

p^{- N} \leq \sum_{n = 1}^{N} p^{- n} .

The tail-sum formula and the assumed exponential intersection tail therefore yield

E X_{r}^{2} \leq \sum_{n = 1}^{\infty} p^{- n} (μ \times μ) (N \geq n) \leq C \sum_{n = 1}^{\infty} (\frac{ζ}{p})^{n} = \frac{Cζ}{p - ζ} .

Solved by gpt-5.6-sol high.

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