Random walks with local memory

TL;DR

This paper establishes a quenched invariance principle for a class of random walks in random environments on d6d, where the walker modifies its environment by updating edges, revealing a stationary environment structure involving the wired uniform spanning forest.

Contribution

It proves a quenched invariance principle for random walks with local environment updates, identifying the stationary environment as a wired uniform spanning forest plus an outgoing edge.

Findings

Proves a quenched invariance principle for these walks.

Identifies the stationary environment as a wired uniform spanning forest.

Shows the environment's structure is invariant under the walk's dynamics.

Abstract

We prove a quenched invariance principle for a class of random walks in random environment on $\mathbb{Z}^d$, where the walker alters its own environment. The environment consists of an outgoing edge from each vertex. The walker updates the edge $e$ at its current location to a new random edge $e'$ (whose law depends on $e$) and then steps to the other endpoint of $e'$. We show that a native environment for these walks (i.e., an environment that is stationary in time from the perspective of the walker) consists of the wired uniform spanning forest oriented toward the walker, plus an independent outgoing edge from the walker.