Browsing by Author "Bazaes, Rodrigo"

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    QUENCHED AND AVERAGED LARGE DEVIATIONS FOR RANDOM WALKS IN RANDOM ENVIRONMENTS: THE IMPACT OF DISORDER
    (2023) Bazaes, Rodrigo; Mukherjee, Chiranjib; Ramirez, Alejandro F.; Saglietti, Santiago
    In 2003, Varadhan (Comm. Pure Appl. Math. 56 (2003) 1222-1245) de-veloped 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 d > 4, 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)), satis-fies 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 ap-proach is the introduction of an auxiliary random walk in a deterministic envi-ronment 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.
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    The effect of disorder on quenched and averaged large deviations for random walks in random environments: Boundary behavior
    (2023) Bazaes, Rodrigo; Mukherjee, Chiranjib; Ramirez, Alejandro F.; Saglietti, Santiago
    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 an := {x is an element of Rd : |x|1 = 1} of their domain which does not intersect any of the (d - 2)-dimensional facets of an, 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 an 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 an). 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 is an element of an there exists ex such that the two rate functions agree at x when the disorder is smaller than ex 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.(c) 2023 Elsevier B.V. All rights reserved.
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    Topics in large deviations and localization for random walks in random environment
    (2021) Bazaes, Rodrigo; Ramírez Chuaqui, Alejandro; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
    En esta tesis se investigó el modelo de "Random Walks in Random Environment" (RWRE). El primer tema en estudiar es sobre la igualdad (o diferencia) entre las funciones de tasa "quenched" y "annealed", en términos del desorden del ambiente. El segundo tema es sobre localización (en la frontera) para RWRE. En particular, se prueba que casi toda distribución de los ambientes es localizada en dimensiones 2 y 3. En ambos problemas, hay una transición de fase para una familia parametrizada de ambientes. En el primer caso, para la igualdad/diferencia entre las funciones de tasa en un punto. En el segundo caso, para deslocalización/localización.