Characterization of planar lattices satisfying the range theorem

Characterize the doubly periodic plane graphs and rotor mechanisms for which recurrence, the deterministic convex limit shape, and the t^{2/3} range law asserted in Theorem 1 hold; in particular, determine whether these conclusions hold for harmonic embeddings of doubly periodic plane graphs.

Background

The paper proves recurrence, a deterministic convex limit shape, and t{2/3} range scaling for the square lattice and for doubly periodic plane graphs of maximum degree three with suitable rotor mechanisms. It also constructs doubly periodic planar counterexamples with attached leaves on which the walk is transient, showing that the theorem does not hold for all planar lattices.

The authors note that harmonic embeddings may form a natural class in which the theorem could remain valid, while simulations indicate that the cyclic rotor order can affect the scaling behavior even on the same graph. The problem therefore asks for a classification of the graph–mechanism pairs for which the theorem’s conclusions hold.

References

For which doubly periodic plane graphs does Theorem~\ref{thm:main} hold? The graph $G_M$ of Proposition~\ref{prop:pendant-counterexample} gives a counterexample in general. However, $G_M$ does not have a harmonic embedding as a plane graph. Recall that an embedding is harmonic when every vertex is the barycenter of its neighbors. We expect Theorem~\ref{thm:main} to hold for harmonic embeddings of doubly periodic plane graphs.

— Eulerian walkers on $\mathbb{Z}^2$ have range exponent $2/3$  (2608.23545 - Bou-Rabee et al., 24 Aug 2026) in Section 7, “Open questions,” Problem 1 (Other planar lattices), labeled prob:planar