Let $\rho \in [0,1]$ and let be $\Lambda$ an accomplishment of the Galton-Watson process with $L=Bin(d, \rho)$ where L provides the distribution of an individual's number of children. Denote by A the event that the origin of the lattice $\Lambda \times \mathbb{Z}.$ Let be the measures quenched and annealed respectively by $$\mathbb{P}_p^{\Lambda\times \mathbb{Z}}(A) \quad ; \quad \mathbb{P}_p^{\rho}(A) = \int_{\Lambda} \mathbb{P}_p^{\Lambda\times \mathbb{Z}}(A) d\Lambda,$$ where $d\Lambda$ is the measure of Galton-Watson process. Define the critical points quenched and annealed respectively by $$p_c^{\Lambda} = \sup \{p : \mathbb{P}_p^{\Lambda\times \mathbb{Z}}(A) = 0\} \quad ; \quad p_c^{\rho} = \sup \{ \mathbb{P}_p^{\rho}(A) = 0\}.$$ I would like to show that $p_c^{\Lambda}=p_c^{\rho}, \; \Lambda-$ almost every point (aep).
I tried to do this by showing that the two inequalities are worth it, but I managed to prove only one side. I would like some tip to prove the other. What I managed to prove was the following. Suppose that $p < p_c^{\rho}.$ Then, $$\int_{\Lambda} \mathbb{P}_p^{\Lambda\times \mathbb{Z}}(A) d\Lambda = 0 \Longrightarrow \mathbb{P}_p^{\Lambda\times \mathbb{Z}}(A) = 0, \Lambda-\textrm{aep} \Longrightarrow p_c^{\Lambda}\geq p_c^{\rho}.$$