Critical bias and recurrent probability laws

Determine whether there is a single critical westward bias p separating recurrence from transience for the clockwise rotor walk on the square lattice, characterize all probability vectors on the four directions that yield recurrence, and determine whether the recurrent set is convex.

Background

For independent initial rotors on the clockwise square lattice, the paper establishes recurrence for distributions sufficiently close to the uniform law and transience for sufficiently strong westward bias. The intermediate regime is not resolved.

The authors further conjecture that the set of recurrent direction-probability vectors is star-shaped about the uniform vector, and explicitly ask whether it is convex.

References

Is there a single critical $p$ separating recurrence from transience? More generally, which probability vectors on the four directions give a recurrent walk? We conjecture that they form a set that is star-shaped about the uniform vector. Is it convex?

— 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 2 (Monotonicity in the bias)