  We consider random walks in Dirichlet environment, introduced by Enriquez and
Sabot in 2006. As this distribution on environments is not uniformly elliptic,
the annealed integrability of exit times out of a given finite subset is a
non-trivial property. We provide here an explicit equivalent condition for this
integrability to happen, on general directed graphs. Such integrability
problems arise for instance from the definition of Kalikow auxiliary random
walk. Using our condition, we prove a refined version of the ballisticity
criterion given by Enriquez and Sabot.
