Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 214 / 1 / b / i

Table of contents
- Solution i

Solution

Let

a_{n} = - lo g P_{p} (0 \leftrightarrow e_{n}) .

{0 \leftrightarrow e_{n}}

and

{e_{n} \leftrightarrow e_{n + m}}

are increasing events of bond percolation. The FKG inequality and translation invariance give

P_{p} (0 \leftrightarrow e_{n + m}) \geq P_{p} (0 \leftrightarrow e_{n}) P_{p} (e_{n} \leftrightarrow e_{n + m}) = P_{p} (0 \leftrightarrow e_{n}) P_{p} (0 \leftrightarrow e_{m}) .

Thus

(a_{n})

is a subadditive sequence. Since every probability is positive for

p > 0

, Fekete lemma applies and gives

ϕ (p) = n \to \infty lim a_{n} / n = n \geq 1 in f [- \frac{1}{n} lo g P_{p} (0 \leftrightarrow e_{n})] .

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