  Consider a random walk in an i.i.d. uniformly elliptic environment in
dimensions larger than one. In 2002, Sznitman introduced for each
$\gamma\in(0,1)$ the ballisticity condition $(T)_{\gamma}$ and the condition
$(T')$ defined as the fulfillment of $(T)_{\gamma}$ for each $\gamma\in(0,1)$.
Sznitman proved that $(T')$ implies a ballistic law of large numbers.
Furthermore, he showed that for all $\gamma\in (0.5,1)$, $(T)_{\gamma}$ is
equivalent to $(T')$. Recently, Berger has proved that in dimensions larger
than three, for each $\gamma\in (0,1)$, condition $(T)_{\gamma}$ implies a
ballistic law of large numbers. On the other hand, Drewitz and Ram\'{{\i}}rez
have shown that in dimensions $d\ge2$ there is a constant
$\gamma_d\in(0.366,0.388)$ such that for each $\gamma\in(\gamma_d,1)$,
condition $(T)_{\gamma}$ is equivalent to $(T')$. Here, for dimensions larger
than three, we extend the previous range of equivalence to all
$\gamma\in(0,1)$. For the proof, the so-called effective criterion of Sznitman
is established employing a sharp estimate for the probability of atypical
quenched exit distributions of the walk leaving certain boxes. In this context,
we also obtain an affirmative answer to a conjecture raised by Sznitman in 2004
concerning these probabilities. A key ingredient for our estimates is the
multiscale method developed recently by Berger.
