Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 212 / 4

Write ${\tau }_p(x)={\mathbb{P}}_p(0⟷x)$ and $r=1-\chi (p{)}^{-1}$. The percolation susceptibility decomposes over the ${\ell }^1$ spheres asIf ${\mathbb{E}}_p{M}_n>r^n$ for every $n\ge 1$, thena contradiction. Hence some $m\ge 1$ satisfies ${\mathbb{E}}_p{M}_m\le r^m$.

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Set $a={\mathbb{E}}_p{M}_m={\sum }_{|x|=m}{\tau }_p(x)$. If $|u|\ge m$, an open self-avoiding path from $0$ to $u$ first meets the ${\ell }^1$ sphere of radius $m$ at some $x$. Its portions from $0$ to $x$ and from $x$ to $u$ use disjoint edge sets. The van den Berg-Kesten inequality and translation invariance therefore giveLet $s_k={\sup }_{|u|\ge km}{\tau }_p(u)$. Since $|u-x|\ge |u|-m$, the last inequality gives $s_k\le a{s}_{k-1}$ and $s_0\le 1$. Induction yields

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For $j=0,\dots ,k-1$, let $E_j$ be the increasing event that $ju⟷(j+1)u$. Their intersection implies $0⟷ku$. By translation invariance and the Harris-FKG inequality,

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Combine parts (a)--(c). For every $k\ge 1$,Taking $k$th roots and letting $k\to \infty$ gives

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