Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 204 / 2 / c / Solution

The open descendants of any vertex form a Galton-Watson process with offspring distribution $\mathrm{Bin}(k,p)$. If $kp\le 1$, it dies out almost surely, so $N_{\infty }=0$.

If $kp>1$, let $\theta >0$ be its survival probability. At level $n$, each vertex begins an independent descendant subtree; the event that its edge to its parent is closed while its descendant open cluster is infinite has probability $(1-p)\theta >0$. Thus the number of such vertices at level $n$ is binomial with $k^n$ trials and a fixed positive success probability. For every fixed $M$, the probability of at least $M$ successes tends to one. Their infinite clusters are separated by their closed parent edges, so $\mathbb{P}(N_{\infty }\ge M)=1$ for every $M$, and hence $N_{\infty }=\infty$ almost surely.