  For a random walk in an elliptic i.i.d. random environment in dimension
greater than or equal to 4, satisfying the a ballisticity condition slightly
weaker than condition (T'), We consider the probability of linear slowdown. We
show an upper bound for this probability which is very close to the lower bound
obtained by the "naive trap" analysis.
  As a tool for obtaining the main result, we show an almost local version of
the quenched central limit theorem under the same ballisticity condition.
