The last few days I am trying my best to understand a part of a proof from these lecture notes on page 14:
The setting is percolation on a regular tree with degree $r$, $C_{BP}(x)$ denotes the cluster of $x$ and $h(\cdot)$ is the distance of the vertex from the origin. It is clear to me why $\theta_n$ satisfies the recursion (1.57).
However, I was not able to show that (1.57) together with $p_c=1/(r-1)$ implies that $\theta_n=(C_\rho+o(1))/n$ for some constant $C_\rho>0$.
After some looking around I have also found this problem as an exercise with the hint to consider $v_n=1/\theta_n$ and performing induction on $n$. Unfortunately I am really stuck and don't see what to do. Maybe someone can help me. Is there hope for an explicit formula for $\theta_n$ or $v_n$? Thank's a lot for your help and time.