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 Zd with d≥4, we show that the quenched and annealed large deviation rate functions agree on any compact set contained in the boundary ∂D≔{x∈Rd:|x|1=1} of their domain which does not intersect any of the (d−2)-dimensional facets of ∂D, provided that the disorder of the environment is low enough (depending on the compact set). As a consequence, we obtain a simple explicit formula for both rate functions on any such compact set of ∂D at low enough 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 ∂D). In addition, our results complement those in Bazaes et al. (2022), 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∈∂D there exists ɛx such that the two rate functions agree at x when the disorder is smaller than ɛx and disagree when it is larger. This further reconfirms the idea, introduced in Bazaes et al. (2022), that the disorder of the environment is in general intimately related with the equality of the rate functions.