  We give new criteria for ballistic behavior of random walks in random
environment which are perturbations of the simple symmetric random walk on
$\mathbb Z^d$ in dimensions $d\ge 4$. Our results extend those of Sznitman
[Ann. Probab. 31, no. 1, 285-322 (2003)] and the recent ones of Ram\'irez and
Saglietti [Preprint, arXiv:1808.01523], and allow us to exhibit new examples in
dimensions $d\ge 4$ of ballistic random walks which do not satisfy Kalikow's
condition. Our criteria implies ballisticity whenever the average of the local
drift of the walk is not too small compared with an appropriate moment of the
centered environment. The proof relies on a concentration inequality of
Boucheron et al. [Ann. Probab. 33, no. 2, 514-560 (2005)].
