  We consider random walk among random conductances where the conductance
environment is shift invariant and ergodic. We study which moment conditions of
the conductances guarantee speed zero of the random walk. We show that if there
exists \alpha>1 such that E[log^\alpha({\omega}_e)]<\infty, then the random
walk has speed zero. On the other hand, for each \alpha>1 we provide examples
of random walks with non-zero speed and random walks for which the limiting
speed does not exist that have E[log^\alpha({\omega}_e)]<\infty.
