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

For a random walk in uniformly elliptic and i.i.d. environment on Zd with d?4, we show that the quenched annealed large deviation rate functions agree any compact set contained boundary ?D?{x?Rd:|x|1=1} of their domain which does not intersect (d?2)-dimensional facets ?D, provided disorder is low enough (depending set). As consequence, obtain simple explicit formula for both such ?D at disorder. In contrast to previous works, our results do assume ballistic behavior are restricted neighborhoods given point (on ?D). addition, complement those Bazaes et al. (2022), where, using different methods, investigate equality interior domain. Finally, general parametrized family environments, strength determines phase transition functions, sense each x??D there exists ?x two x when smaller than disagree it larger. This further reconfirms idea, introduced intimately related functions.