  In 2003, Varadhan [V03] developed a robust method for proving quenched and
averaged large deviations for random walks in a uniformly elliptic and i.i.d.
environment (RWRE) on $\mathbb Z^d$. One fundamental question which remained
open was to determine when the quenched and averaged large deviation rate
functions agree, and when they do not. In this article we show that for RWRE in
uniformly elliptic and i.i.d. environment in $d\geq 4$, the two rate functions
agree on any compact set contained in the interior of their domain which does
not contain the origin, provided that the disorder of the environment is
sufficiently low. Our result provides a new formulation which encompasses a set
of sufficient conditions under which these rate functions agree without
assuming that the RWRE is ballistic (see [Y11]), satisfies a CLT or even a law
of large numbers ([Zer02,Ber08]). Also, the equality of rate functions is not
restricted to neighborhoods around given points, as long as the disorder of the
environment is kept low. One of the novelties of our approach is the
introduction of an auxiliary random walk in a deterministic environment which
is itself ballistic (regardless of the actual RWRE behavior) and whose large
deviation properties approximate those of the original RWRE in a robust manner,
even if the original RWRE is not ballistic itself.
