Write $\tau_p(x)=\mathbb P_p(0\longleftrightarrow x)$ and $r=1-\chi(p)^{-1}$. The <percolation susceptibility> decomposes over the $\ell^1$ spheres as
$$
\chi(p)=\mathbb E_p|C|
=\sum_{n=0}^\infty\mathbb E_pM_n,
\qquad \mathbb E_pM_0=1.
$$
If $\mathbb E_pM_n>r^n$ for every $n\geq1$, then
$$
\chi(p)>1+\sum_{n=1}^\infty r^n
=1+\frac r{1-r}=\chi(p),
$$
a contradiction. Hence some $m\geq1$ satisfies $\mathbb E_pM_m\leq r^m$.

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-212/4/a/solution