June 2023 Quenched and averaged large deviations for random walks in random environments: The impact of disorder
Rodrigo Bazaes, Chiranjib Mukherjee, Alejandro F. Ramírez, Santiago Saglietti
Author Affiliations +
Ann. Appl. Probab. 33(3): 2210-2246 (June 2023). DOI: 10.1214/22-AAP1864


In 2003, Varadhan (Comm. Pure Appl. Math. 56 (2003) 1222–1245) developed a robust method for proving quenched and averaged large deviations for random walks in a uniformly elliptic and i.i.d. environment (RWRE) on Zd. One fundamental question which remained open was to determine when the quenched and averaged large deviation rate functions agree, and when they do not. In this article we show that for RWRE in uniformly elliptic and i.i.d. environment in d4, the two rate functions agree on any compact set contained in the interior of their domain which does not contain the origin, provided that the disorder of the environment is sufficiently low. Our result provides a new formulation which encompasses a set of sufficient conditions under which these rate functions agree without assuming that the RWRE is ballistic (see (Probab. Theory Related Fields 149 (2011) 463–491)), satisfies a CLT or even a law of large numbers (Electron. Commun. Probab. 7 (2002)191–197; Ann. Probab. 36 (2008) 728–738). Also, the equality of rate functions is not restricted to neighborhoods around given points, as long as the disorder of the environment is kept low. One of the novelties of our approach is the introduction of an auxiliary random walk in a deterministic environment which is itself ballistic (regardless of the actual RWRE behavior) and whose large deviation properties approximate those of the original RWRE in a robust manner, even if the original RWRE is not ballistic itself.

Funding Statement

The first author has been supported by ANID-PFCHA/Doctorado Nacional no. 2018-21180873.
The second author is supported by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure.
The third author has been partially supported by Fondo Nacional de Desarrollo Científico y Tecnológico 1180259 and 1220396, Iniciativa Científica Milenio and by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai.
The fourth author has been supported in part at the Technion by a fellowship from the Lady Davis Foundation, the Israeli Science Foundation grants no. 1723/14 and 765/18, the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai and the FONDECYT Iniciación grant no. 11200690.
This research was also supported by a grant from the United States-Israel Binational Science Foundation (BSF), no. 2018330.


The authors are very grateful to Noam Berger, Nina Gantert and Atilla Yilmaz for very useful comments on an earlier version of the manuscript.


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Rodrigo Bazaes. Chiranjib Mukherjee. Alejandro F. Ramírez. Santiago Saglietti. "Quenched and averaged large deviations for random walks in random environments: The impact of disorder." Ann. Appl. Probab. 33 (3) 2210 - 2246, June 2023. https://doi.org/10.1214/22-AAP1864


Received: 1 November 2021; Revised: 1 March 2022; Published: June 2023
First available in Project Euclid: 2 May 2023

MathSciNet: MR4583669
zbMATH: 1511.60151
Digital Object Identifier: 10.1214/22-AAP1864

Primary: 60F10 , 60K37 , 82C41

Keywords: Disorder , Disordered media , large deviations , quenched and averaged rate functions , random walks in random environment , random walks in random scenery

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 3 • June 2023
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