2021/01/12 by Rodrigo Bazaes, Chiranjib Mukherjee, Bazaes, Rodrigo +6
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR
paper · pdf · doi:10.48550/arxiv.2101.04606
arXiv admin note: text overlap with arXiv:1906.05328
openalex publication_date 2021/01/12 · arxiv created 2021/01/31 · arxiv updated 2021/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a random walk in a uniformly elliptic and i.i.d. environment on \mathbb Zd with d ≥ 4, we show that the quenched and annealed large deviations rate functions agree on any compact set contained in the boundary ∂ \mathbbD:=\ x ∈ \mathbb Rd : |x|1 =1\ of their domain which does not intersect any of the (d-2)-dimensional facets of ∂ \mathbbD, provided that the disorder of the environment is~low~enough. As a consequence, we obtain a simple explicit formula for both rate functions on ∂ \mathbbD 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 ∂ \mathbbD). 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 ∈ ∂ \mathbbD there exists εx such that the two rate functions agree at x when the disorder is smaller than εx 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.