Electronic Journal of Probability

Scaling limit of sub-ballistic 1D random walk among biased conductances: a story of wells and walls

Quentin Berger and Michele Salvi

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We consider a one-dimensional random walk among biased i.i.d. conductances, in the case where the random walk is transient but sub-ballistic: this occurs when the conductances have a heavy-tail at $+\infty $ or at $0$. We prove that the scaling limit of the process is the inverse of an $\alpha $-stable subordinator, which indicates an aging phenomenon, expressed in terms of the generalized arcsine law. In analogy with the case of an i.i.d. random environment studied in details in [ESZ09a, ESTZ13], some “traps” are responsible for the slowdown of the random walk. However, the phenomenology is somehow different (and richer) here. In particular, three types of traps may occur, depending on the fine properties of the tails of the conductances: (i) a very large conductance (a well in the potential); (ii) a very small conductance (a wall in the potential); (iii) the combination of a large conductance followed shortly after by a small conductance (a well-and-wall in the potential).

Article information

Electron. J. Probab., Volume 25 (2020), paper no. 30, 43 pp.

Received: 15 April 2019
Accepted: 31 January 2020
First available in Project Euclid: 24 February 2020

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Digital Object Identifier

Primary: 60K37: Processes in random environments 60F17: Functional limit theorems; invariance principles 60G52: Stable processes
Secondary: 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

random walk random environment conductance model sub-ballisticity scaling limit subordinator trapping phenomenon aging localization

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Berger, Quentin; Salvi, Michele. Scaling limit of sub-ballistic 1D random walk among biased conductances: a story of wells and walls. Electron. J. Probab. 25 (2020), paper no. 30, 43 pp. doi:10.1214/20-EJP427. https://projecteuclid.org/euclid.ejp/1582534895

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