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For a random walk in a uniformly elliptic and i.i.d. environment on $\mathbbZ^d$ with $d \geq 4$, we show that the quenched and annealed large deviationsrate 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 notintersect 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 notassume any ballistic behavior of the random walk and are not restricted toneighborhoods of any given point (on the boundary $\partial \mathbb{D}$). Inaddition, our~results complement those in [BMRS19], where, using differentmethods, we investigate the equality of the rate functions in the interior oftheir domain. Finally, for a general parametrized family of environments,we~show that the strength of disorder determines a phase transition in theequality 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 agreeat $x$ when the disorder is smaller than $\varepsilon_x$ and disagree when itslarger. This further reconfirms the idea, introduced in [BMRS19], that thedisorder of the environment is in general intimately related with the equalityof the rate functions.

The effect of disorder on quenched and averaged large deviations for random walks in random environments: boundary behavior