Return Probability for the Lamplighter Walk on a Tree

Precise statement

For the switch-walk-switch lamplighter walk on $\mathbb{Z}_2 \wr T_d$, prove the sharp asymptotic $p_{2n}(e,e) = \rho_d^{2n} \exp[-(\pi^2 (\log(d-1))^2 + o(1)) \frac{n}{\log^2 n}]$ with $\rho_d = \frac{2\sqrt{d-1}}{d}$.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

QED (GPT-5.5 Pro)

The QED multi-agent system produced the proof from the problem statement alone, through multiple rounds of decomposition and refinement.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

independent expert verified

Verified by the contributing domain expert who posed the problem; public preprint.

Correctness: supported · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-05-15

   arXiv:2605.21744 - Return probability for the switch-walk-switch lamplighter walk

   VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.