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

Rodrigo Bazaes, Chiranjib Mukherjee, Alejandro F. Ramírez and Santiago Saglietti

Stochastic Processes and their Applications, 2023, vol. 158, issue C, 208-237

Abstract: 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.

Keywords: random walk in random environment; large deviations (search for similar items in EconPapers)
Date: 2023
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304414923000030
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:158:y:2023:i:c:p:208-237

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.spa.2023.01.003

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().