We are interested in the recurrence and transience of a branching random walk in Z^d indexed by a critical Galton-Watson tree conditioned to survive. When the environment is homogeneous, deterministic, and if the offspring distribution has a second moment, it is known to be recurrent for d at most 4, and transient for d larger than 4. In this talk we consider a random environment made of conductances, and we prove that, if the conductances satisfy suitable assumptions, the same result holds. The argument is based on the combination of a 0-1 law and a truncated second moment method, which only requires to have good estimates on the quenched Green's function and heat kernel of a (non-branching) random walk in random conductances. This is a joint work with Christophe Sabot and Bruno Schapira.
