Roundness of the square-lattice limit shape

Determine whether the limit shape B for the clockwise rotor walk with independent uniform initial rotors on the square lattice is a Euclidean disk.

Background

The main theorem establishes that the rescaled range converges to a deterministic compact convex set B, but does not identify the geometry of B. Numerical simulations reported in the paper suggest rotational symmetry on the square lattice and also indicate disk-shaped limits for several other lattices.

The unresolved question concerns whether the square-lattice limit shape is exactly Euclidean rather than merely convex.

References

Is the limit shape $B$ in Theorem~\ref{thm:main} a Euclidean disk on the square lattice? \citet{KapriDhar2009} conjectured this via simulation.

— 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 3 (Roundness of the limit shape)

For clockwise planar rotors with independent uniform initial states, \citet*{BouRabeePeres2026} proved convergence of the rescaled range to a deterministic compact convex set; whether it is a disk remains open.

— Kozma's centrally excited walk converges to a Euclidean ball  (2610.10162 - Bou-Rabee et al., 7 Oct 2026) in Section 1, Related work (paragraph discussing Eulerian and rotor walks)