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

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.