Recurrence and Diffusion in Alternating-Drift Walks

Definitions and mathematical statement

The source poses model-specific questions for the $H$-walk, $Y$-walk, and two-walker Slugs model.

1. For the $H$-walk on ${\mathbb{Z}}^d$, whose cookie-induced drift alternates between left and right, determine whether it is recurrent for $d=2$ and transient for $d=3$.
2. Determine the analogous recurrence/transience behavior for the $Y$-walk, retaining the comparison with dimensions two and three rather than asserting an answer.
3. In dimension two, determine whether the two-walker Slugs model is diffusive and transient.

For a position process $(X_m)$ started at the origin, a standard recurrence assertion is $\Pr(X_m=0\text{ infinitely often})=1$. A standard transience assertion requires finitely many visits to each fixed site almost surely. For non-Markovian walks, the relevant starting state, environment, probability law, and any zero--one statement must be specified rather than assumed from terminology.

The record does not give the actual transition probabilities or cookie rule for the $H$-walk: alternating drift could depend on local visit number, time, or another state variable, and its strength and initial phase are missing. The $Y$-walk is only named. For Slugs, the interaction, update schedule, initial positions, and which observable is tested for recurrence or transience (each walker or the pair) are not supplied.

Diffusivity also needs a precise target. Growth of mean-square displacement proportional to time, convergence of positions after square-root rescaling, and a functional Brownian limit are different statements; no one is chosen without the source's definition. The walk clock and any centering must be given. The source's requested two-walker transience is not silently replaced by transience of an unspecified center-of-mass process.

The cited periodic-cookie work is contextual. It cannot define the three named models or resolve their exact questions without an explicit identification of transition rules and hypotheses.

Short English statement

The entry asks about recurrence or transience for two named alternating-drift walks and about diffusion and transience for a two-walker model in the plane. Their transition rules and the precise diffusion and two-walker observables are missing.

Sources

• Local UnsolvedMath registry, permanent ID 30000772, code catalogue entry, "Recurrence and Diffusion in Alternating-Drift Walks." Proposed year 2007; proposer unspecified.
• Non-Classical Interacting Random Walks (2008), Oberwolfach Reports, pp. 1521--1574, DOI 10.4171/owr/2007/27; https://doi.org/10.4171/owr/2007/27. The displayed report year 2008 differs from the DOI and proposed year 2007.
• Supplied "Kozma--Orenshtein--Shinkar, periodic cookie walks": https://arxiv.org/abs/1311.7439.
• Statement status corrected_verified; the supplied note says official-report content was checked and wording/notation normalized without changing the claim.
• Literature triage checked 2026-08-21: open. It says the three exact models retain model-specific recurrence, transience, and scaling questions not resolved by general excited-walk theory. Exact models and current status require independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying report and literature provenance remain as supplied. Unreviewed statement-only draft.