Scaling Limits of Line Models in Degenerate Environment

TL;DR

This paper analyzes a 2D random walk in a degenerate environment, revealing superdiffusive behavior in heavy-tailed directions and providing conditions for non-explosion, with explicit scaling limits and conjectures for complex cases.

Contribution

It introduces a detailed analysis of line models in degenerate environments, including explicit scaling limits and conditions for non-explosion, advancing understanding of such stochastic processes.

Findings

Superdiffusive behavior in heavy-tailed directions.

Explicit scaling limit as a time-changed Brownian motion.

Conditions for non-explosion in degenerate environments.

Abstract

We consider a 2-dimensional model of random walk in random environment known as line model. The environment is described by two independent families of i.i.d. random variables dictating rates of jumps in vertical, respectively horizontal directions, and whose values are constant along vertical, respect. horizontal lines. When jump rates are heavy-tailed in one of the directions, we prove that the random walk becomes superdiffusive in that direction, with an explicit scaling limit written as a two-dimensional Brownian motion time-changed (in one of the components) by a process introduced by Kesten and Spitzer in 1979. In the case of a fully degenerate environment, a sufficient condition for non-explosion is provided, and conjectures on the associated scaling limit are formulated.