Eulerian walkers on $\mathbb{Z}^2$ have range exponent $2/3$

What the math gives to ML

The paper provides an operational deterministic analogue of random-walk exploration: local cyclic rotor updates generate a globally constrained trajectory, and the circuit map gives an order-independent way to expand explored sets. The transferable asset is not the specific t^{1/3} exponent, which is tied to the square lattice, but the abelian-style routing and bounded-discrepancy visitation induced by cyclic local state. This can replace independently restarted random-walk neighborhood sampling in graph neural networks with a stateful sampler that systematically covers neighborhoods while preserving a controllable exploration budget. The main test is whether this reduces minibatch sampling variance and improves validation accuracy at equal sampled edges, rather than merely producing a different traversal.