Random Eulerian excursion dichotomy on high-dimensional tori

Definitions and mathematical statement

Fix $d\ge3$. For each $N\ge3$, form the nearest-neighbor torus $(\mathbb Z/N\mathbb Z)^d$ and replace each undirected edge by its two opposite directed arcs. Choose uniformly an Eulerian circuit, meaning a closed traversal using every directed arc exactly once. Root its traversal at the origin; choosing uniformly among origin-rooted traversals gives the same law for the unordered excursion lengths.

Cut the circuit at successive returns to the origin. There are $2d$ excursions, and their lengths, measured in directed steps, sum to $2dN^d$. For a deterministic threshold $\omega_N\to\infty$ with $\omega_N^2<N^d$, let $b_N$ count lengths greater than $N^d/\omega_N$, let $t_N$ count lengths smaller than $\omega_N$, and let $m_N$ count the remaining lengths, including any endpoint equalities.

Is there a probability law $S^*$ on $\{1,\ldots,2d\}$ such that, for thresholds tending to infinity sufficiently slowly, \[ (b_N,t_N,m_N)\xrightarrow{d}(S^*,2d-S^*,0)? \] One precise meaning of "sufficiently slowly" is that some deterministic $a_N\to\infty$, with $a_N=o(N^{d/2})$, works for every divergent $\omega_N\le a_N$ eventually.

Convergence in distribution here is convergence of probabilities on the finite set of integer triples. The limiting law may depend on $d$, but not on the chosen slow threshold. The target asserts that intermediate-sized excursions disappear and that a positive number remain on the macroscopic scale. It does not prescribe the exact law of the macroscopic lengths themselves.

Short English statement

Do random Eulerian excursions on higher-dimensional tori split into microscopic and macroscopic groups with no intermediate lengths?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 9700013 (AMR-096-0013), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Aldous - Open problems, Random Eulerian Circuits, Conjecture 0.1. Original source indexed by AMR-096-0013. https://www.stat.berkeley.edu/~aldous/Research/OP/index.html.
• S3 D. Aldous and J. Yu, Random Eulerian Circuits (2014), original numbered conjecture and examples. https://www.stat.berkeley.edu/~aldous/Papers/aldous_eulerian.pdf.