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Scaling limit of sub-ballistic 1D random walk among biased conductances: a story of wells and walls. (English) Zbl 1441.60090
Summary: 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 [N. Enriquez et al., Bull. Soc. Math. Fr. 137, No. 3, 423–452 (2009; Zbl 1186.60108); Ann. Appl. Probab. 23, No. 3, 1148–1187 (2013; Zbl 1279.60126)], 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).
Reviewer: Reviewer (Berlin)
60K37 Processes in random environments
60F17 Functional limit theorems; invariance principles
60G52 Stable stochastic processes
82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
Full Text: DOI Euclid
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