Sharp transition in self-avoiding walk on random conductors on a tree

We consider self-avoiding walk on random conductors on the degree-$ell$ tree. It is known that there exists a transition behavior between the strong and the weak disorder regime. We prove that in weak disorder regimes, the quenched critical point is equal to the annealed one, and the critical exponent for the quenched susceptibility takes on the mean field value 1. We also prove by estimating the fractional moment that the quenched critical point is strictly smaller than the annealed one in the strong disorder regime. Moreover, we obtain a lower bound of the difference between those two critical points quantatively.