Let
$$
a_n=-\log\mathbb P_p(0\leftrightarrow e_n).
$$
The events $\{0\leftrightarrow e_n\}$ and $\{e_n\leftrightarrow e_{n+m}\}$ are increasing events of <bond percolation>. The <FKG inequality> and translation invariance give
$$
\mathbb P_p(0\leftrightarrow e_{n+m})
\geq \mathbb P_p(0\leftrightarrow e_n)
\mathbb P_p(e_n\leftrightarrow e_{n+m})
=\mathbb P_p(0\leftrightarrow e_n)
\mathbb P_p(0\leftrightarrow e_m).
$$
Thus $(a_n)$ is a <subadditive sequence>. Since every probability is positive for $p>0$, <Fekete lemma> applies and gives
$$
\boxed{\phi(p)=\lim_{n\to\infty}a_n/n
=\inf_{n\geq1}\left[-\frac1n\log\mathbb P_p(0\leftrightarrow e_n)\right].}
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-214/1/b/i/solution