Abstract
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 [Enriquez, Sabot, Zindy, Bull. Soc. Math. 2009; Enriquez, Sabot, Tournier, Zindy, Ann. Appl. Probab. 2013], 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).
Highlights
Random walks in random environment have been studied extensively over the past forty years, both from a physical and a mathematical perspective
The behavior of recurrent random walks in random environment has been further studied by Sinai in [Sin82], who showed a strong slowdown of the walk, with an unusual scaling2
The case of transient sub-ballistic random walks in i.i.d. random environment has been considered in [KKS75], and in [ESZ09a, ESZ09b, ESTZ13]
Summary
Random walks in random environment have been studied extensively over the past forty years, both from a physical and a mathematical perspective. Random environment, conductance model, sub-ballisticity, scaling limit, subordinator, trapping phenomenon, aging, localization. Before showing how to derive Theorem 1.2 from Theorem 2.1, we need the following proposition, which slightly improves Proposition 2.1 in [BS19]: we prove that the walk (Xj)j≥0 cannot backtrack more than C log n before reaching distance n We will use this property several times in the proof of the main theorem, but it already tells us that the map n → Tn is the inverse of the map j → Xj, up to an error of at most a constant times log n.
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