Eulerian walker range conjecture of Priezzhev, Dhar, Dhar, and Krishnamurthy

Let each site of $\mathbb{Z}^2$ be assigned an independent uniformly random initial direction among its four outgoing directions. Starting at the origin, the Eulerian walker turns the arrow at its current site clockwise by $90^\circ$ and then moves along the resulting arrow. If $R_t$ denotes the radius of the set of sites visited during the first $t$ steps, then the conjecture asserts that $R_t$ has order $t^{1/3}$ (equivalently, the visited range has exponent $2/3$ ).

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Range-exponent formulation
For the same Eulerian walker with independent uniformly random initial directions, the number of lattice sites visited during the first $t$ steps has growth exponent $2/3$ ; equivalently, the range is of order $t^{2/3}$ in the sense asserted by the conjecture.
source: Eulerian walkers on $\mathbb{Z}^2$ have range exponent $2/3}

References

Primary source

arXiv

Additional references

Eulerian walkers on Z^2 have range exponent 2/3 — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new preprint claims to settle the conjecture, but its result has not yet been independently verified.

The conjecture of Priezzhev, Dhar, Dhar, and Krishnamurthy concerns the long-term size, recurrence, and shape of the region visited by a two-dimensional Eulerian walker.

Known results

Earlier work argued heuristically and numerically that the visited radius grows as $N^{1/3}$ , so the range scales as $N^{2/3}$ .
Simulations found an approximately circular rescaled region and estimated boundary-width exponent $\alpha=0.40\pm0.06$ .
On graphs without endpoints, the dynamics was shown to enter a Poincaré cycle; this does not establish the conjectured range or limiting shape.
The available literature treated these conclusions as heuristic or numerical evidence, not a proof.

August 24, 2026 claimed proof

On August 24, 2026, the preprint Eulerian walkers on $\mathbb{Z}^2$ have range exponent $2/3$ claimed the conjectured range exponent, infinite recurrence, and convergence of the rescaled visited region. This is a complete-resolution claim, but it is unrefereed and remains unverified.

Current status (as of August 2026): The conjecture has a new unrefereed preprint claiming all stated conclusions, but no independently verified proof is recorded.

Sources

No solutions have been posted yet.

Range-exponent formulation

Primary source

Additional references

Progress summary