Diffusive scaling in dimensions greater than two

Determine whether the rotor walk on the d-dimensional lattice, for a fixed translation-invariant rotor mechanism and independent uniform initial rotors with d>2, converges to Brownian motion under diffusive scaling.

Background

The paper establishes recurrence and t{2/3} range behavior in selected two-dimensional settings, while the corresponding large-scale behavior in higher dimensions remains unresolved. The question concerns a fixed translation-invariant rotor mechanism on the d-dimensional lattice with independent uniform initial directions.

A positive answer would establish a Brownian-motion scaling limit and would formalize the earlier prediction that rotor walks have diffusive mean-square displacement in every dimension greater than two.

References

For a fixed translation-invariant rotor mechanism on $d$ with $d>2$ and independent uniform initial rotors, does the walk converge to Brownian motion under diffusive scaling? \citet{PriezzhevDharDharKrishnamurthy1996} predicted diffusive mean-square displacement in every dimension $d>2$.

— 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 4 (Diffusive scaling in higher dimensions)