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

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

Solved by gpt-5.6-sol high.