Return Probability for the Lamplighter Walk on a Tree

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 $\rho_d = \frac{2\sqrt{d-1}}{d}$.

Result

Proved

Status

Resolved

AI contribution

AI-discovered

Method

Argument

Field

Probability on groups

Posed by

—

Year posed

2025

Years open

1y

Solved

2026-05-15

Model

QED (GPT-5.5 Pro)

Vendor

OpenAI

Collaborators

—

Verification

Independently expert-verified

Publication

Preprint

Significance

10 / 100*

Disclosed cost

—

Wikipedia

No dedicated article