Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 204 / 2 / f / Solution

Use the standard theorem that supercritical Bernoulli percolation on ${\mathbb{Z}}^3$ has a unique infinite open cluster almost surely. Fix $p>p_c$ and choose $q\in (p_c,p)$. Almost surely ${\eta }_q$ has an infinite cluster somewhere. On $I_p$, the origin and that cluster lie in the unique infinite ${\eta }_p$-cluster, so a finite ${\eta }_p$-open path joins them. Almost surely every label on this finite path is strictly below $p$; increasing $q$ to some $r<p$ above those finitely many labels makes the origin percolate in ${\eta }_r$. Hence $I_p\subseteq \{M<p\}$ up to a null event, and part e gives left continuity at $p$. Together with part c, $\theta$ is continuous on $(p_c,1]$.