# Scaling Limit of Sub-ballistic 1D Random Walk among Biased Conductances: a Story of Wells and Walls

## TL;DR

This paper studies the scaling limit of a one-dimensional biased random walk with heavy-tailed conductances, revealing an inverse stable subordinator limit and a complex trapping phenomenology involving wells and walls.

## Contribution

It establishes the inverse $ ext{α}$-stable subordinator as the scaling limit for sub-ballistic biased walks with heavy-tailed conductances, highlighting new trap types and aging effects.

## Key findings

- Scaling limit is an inverse α-stable subordinator.
- Identification of three trap types: wells, walls, and combined traps.
- Demonstration of aging phenomena via generalized arcsine law.

## 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).