Scaling Limits of Line Models in Degenerate Environment

TL;DR

This work analyzes a two-dimensional line-conductance random walk in a degenerate random environment, where horizontal and vertical jump rates are constant along lines and follow heavy-tailed distributions. It proves a non-explosion criterion in terms of tail exponents and establishes scaling limits: when both tails are light, a standard quenched invariance principle to Brownian motion holds; in semi-degenerate regimes, the horizontal coordinate becomes a Brownian motion time-changed by a Kesten–Spitzer clock driven by the vertical Brownian motion, with a parallel result for the constant-speed variant involving the KS clock inverse. The analysis hinges on time-change techniques that decouple coordinates, martingale convergence via Girsanov arguments, and convergence of local times, culminating in a detailed KS-type limit description. These results illuminate how heavy-tailed, line-structured environments yield anisotropic, non-Gaussian scaling and highlight the role of random-scenery type limits in diffusion in random media.

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.

Figures (1)

• Figure 1: Trajectory of line model for $\alpha_1=0.6$, $\alpha_2=0.9$, after 100 jumps

Theorems & Definitions (44)

• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Proposition 3.1
• Remark 3.2
• Lemma 3.3
• proof : Proof of Lemma \ref{['lem:Xstar_is_conservative']}
• Proposition 3.4
• Remark 3.5