On the connection between transient and ballistic behaviours for RWRE

We study the strong form of the ballistic conjecture for random walks in random environments (RWRE). This conjecture asserts that any RWRE which is directionally transient for a nonempty open set of directions satisfies condition $(T)$ (annealed exponential decay for the unlikely exit probability). Specifically, we introduce a ballisticity condition which is fulfilled as soon as a polynomial condition of degree greater than $d-1$ holds. Under that hypothesis we prove condition $(T)$, which turns this condition into the weakest-known ballisticity assumption. We recall that standard arguments to prove that a ballisticity condition implies directional transience require at least polynomial decay greater than degree $d$. Furthermore, in the one dimensional case we provide an alternative proof which proves the equivalence between transient behaviour and annealed arbitrary decay for the unlikely exit probability, we expect that this new argument might be used in higher dimensions.