Random Walks in Random Environments with Rare Anomalies

TL;DR

This work investigates random walks in random environments on $ abla ext{Z}^d$ with blue and red site types, introducing a rare anomaly framework in which rare red sites still influence ballistic behavior. The authors develop a coupling-based, truncation-driven approach that compares blue and red local times via truncated Green's functions to obtain explicit velocity bounds, recovering Kalikow's ballisticity results with quantifiable constants. In the rare anomaly regime, they show that when red sites satisfy two-direction ellipticity and occur with sufficiently small density, the asymptotic velocity can be bounded arbitrarily close to the blue-site drift, with a bound depending only on the ellipticity parameter and not on the red-site distribution. The paper also demonstrates the necessity of the i.i.d. assumption using a strongly mixing counterexample and discusses open questions about relaxing assumptions and extending the results to broader ballisticity criteria.

Abstract

We study random walks in i.i.d. random environments on $\mathbb{Z}^d$ when there are two basic types of vertices, which we call "blue" and "red". Each color represents a different probability distribution on transition probability vectors. We introduce a method of studying these walks that compares the expected amount of time spent at a specific site on the event that the site is red with the expected amount of time spent there on the event that the site is blue. This method produces explicit bounds on the asymptotic velocity of the walk. We recover an early result of Kalikow, but with new bounds on the velocity. Next, we consider a "rare anomaly" model where the vast majority of sites are blue, and blue sites are uniformly elliptic, with some almost-sure bounds on the quenched drift. We show that if the red sites satisfy a certain uniform ellipticity assumption in two fixed, non-parallel directions, then even if red sites break the almost-sure bounds on the quenched drift, making red sites unlikely enough lets us obtain bounds on the asymptotic velocity of the walk arbitrarily close to the bounds on the quenced drift at blue sites. Significantly, the required proportion $p^*$ of blue sites to do this does not depend on the distribution of red sites, except through the uniform ellipticity assumption in two directions. Our proof is based on a coupling technique, where two walks run in environments that are the same everywhere except at one vertex. They decouple when they hit that vertex, and our proof is driven by bounds on how long it takes to recouple. We then demonstrate the importance of the i.i.d. assumption by providing a counterexample to the statement of the theorem with this assumption removed. We conclude with open questions.

Figures (4)

• Figure 1: Kalikow's two-vertex model is depicted on the top. On the bottom is depicted the two-vertex model with the specific values Kalikow presented at the beginnng of Kalikow1981.
• Figure 2: The case described in Theorem \ref{['thm:SpecialCase']}
• Figure 3: The set $\mathcal{S}$ is depicted when $d=2$ on the left, and when $d=3$ on the right using an orthographic rendering. The origin, which is not in $\mathcal{S}$, is black, and vertices in $\mathcal{S}$ are purple.
• Figure 4: All nearest neighbors of $y$ (circled in orange) can be reached via paths of length no more than 5, where each step has probability at least $\kappa$ in $\omega_3$. The situations when $d=2$ and $d=3$ are depicted on the left and right, respectively.

Theorems & Definitions (54)

• Theorem 1
• Theorem 2
• Remark
• Theorem 3
• Remark
• Example 1
• Example 2: Nearly non-nestling
• Example 3: Double perturbation of a zero-drift simple random walk
• Proposition
• Corollary 4