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

PutFix $k\ge 1$ and write $n=qk+r$ with $0\le r<k$. Repeated subadditivity givessoThe finitely many values $x_0,\dots ,x_{k-1}$ are bounded, henceTaking the infimum over $k$ gives $\mathrm{limsup}{x}_n/n\le \alpha$, while the definition of $\alpha$ gives $x_n/n\ge \alpha$ for every $n$. ThereforeThis is Fekete lemma.

LetThe events $\{0\leftrightarrow e_n\}$ and $\{e_n\leftrightarrow e_{n+m}\}$ are increasing events of bond percolation. The FKG inequality and translation invariance giveThus $(a_n)$ is a subadditive sequence. Since every probability is positive for $p>0$, Fekete lemma applies and gives

By the union bound,Hence some $z\in \partial B(n)$ has connection probability at least the left side divided by $|\partial B(n)|$. Some coordinate of $z$ equals $n$ or $-n$. A coordinate permutation and reflection of ${\mathbb{Z}}^d$ sends that face to the face $x_1=n$ and preserves the bond percolation law. Its image $x$ therefore satisfies

WriteSince $e_n\in \partial B(n)$, $q_n\ge a_n$, and thereforeFor the opposite inequality, choose $x=(n,x_2,\dots ,x_d)$ as in part (ii). Reflection about $x$ sends $0$ to $2n{e}_1$ and preserves the lattice. Thus the increasing events $\{0\leftrightarrow x\}$ and $\{x\leftrightarrow 2n{e}_1\}$ have the same probability. The FKG inequality yieldsConsequentlyThe polynomial boundary-size estimate makes the last term tend to zero, while the first tends to $\varphi (p)$. Hence