Random walk in mixed random environment without uniform ellipticity

TL;DR

This paper investigates a random walk in a mixed environment on non-negative integers, analyzing how the interplay of heavy-tailed sites and fast points influences recurrence, transience, and trajectory bounds, revealing phase transitions based on model parameters.

Contribution

It introduces a model combining heavy-tailed and fast points in a non-uniform environment and characterizes the conditions for recurrence, transience, and phase transitions.

Findings

Identifies phase transitions based on environment parameters.

Provides conditions for recurrence and transience.

Establishes almost-sure bounds for the walk's trajectories.

Abstract

We study a random walk in random environment on the non-negative integers. The random environment is not homogeneous in law, but is a mixture of two kinds of site, one in asymptotically vanishing proportion. The two kinds of site are (i) points endowed with probabilities drawn from a symmetric distribution with heavy tails at 0 and 1, and (ii) `fast points' with a fixed systematic drift. Without these fast points, the model is related to the diffusion in heavy-tailed (`stable') random potential studied by Schumacher and Singh; the fast points perturb that model. The two components compete to determine the behaviour of the random walk; we identify phase transitions in terms of the model parameters. We give conditions for recurrence and transience, and prove almost-sure bounds for the trajectories of the walk.