  For a random walk in a uniformly elliptic and i.i.d. environment on $\mathbb
Z^d$ with $d \geq 4$, we show that the quenched and annealed large deviations
rate functions agree on any compact set contained in the boundary $\partial
\mathbb{D}:=\{ x \in \mathbb R^d : |x|_1 =1\}$ of their domain which does not
intersect any of the $(d-2)$-dimensional facets of $\partial \mathbb{D}$,
provided that the disorder of the environment is~low~enough. As a consequence,
we obtain a simple explicit formula for both rate functions on $\partial
\mathbb{D}$ at low disorder. In contrast to previous works, our results do not
assume any ballistic behavior of the random walk and are not restricted to
neighborhoods of any given point (on the boundary $\partial \mathbb{D}$). In
addition, our~results complement those in [BMRS19], where, using different
methods, we investigate the equality of the rate functions in the interior of
their domain. Finally, for a general parametrized family of environments,
we~show that the strength of disorder determines a phase transition in the
equality of both rate functions, in the sense that for each $x \in \partial
\mathbb{D}$ there exists $\varepsilon_x$ such that the two rate functions agree
at $x$ when the disorder is smaller than $\varepsilon_x$ and disagree when its
larger. This further reconfirms the idea, introduced in [BMRS19], that the
disorder of the environment is in general intimately related with the equality
of the rate functions.
