Total-variation decay for lamplighter walks on the integer line

The question is whether for the discrete-time switch–walk–switch walk on Z₂ wr Z, started with all lamps off and walker positions 0 or 2, the total-variation distance between the time-t laws is comparable to t^(−1/2), with constants independent of t.

Why it matters and what counts as progress

Why it matters: It determines how quickly the specified two lamplighter starting states become statistically indistinguishable, with matching upper and lower rates.

What counts as progress: Obtain matching upper and lower total-variation bounds for the specified starting states and discrete-time walk.

How the problem developed

15 May 2026 · Reported result [1]
QED and a domain expert A QED project record gives a proof that switch–walk–switch laws from the all-off states at positions 0 and 2 have total-variation distance comparable to t⁻¹ᐟ²; the domain expert confirmed the proof.
26 Jun 2026 · Public report [2]
Chenyang An, Qihao Ye, Minghao Pan and Jiayun Zhang Published the exact start-state comparison and asymptotic in the QED system paper, documenting the expert-confirmed result.

AI contribution

QED: An Open-Source Multi-Agent System for Generating Mathematical Proofs on Open Problems

Chenyang An, Qihao Ye, Minghao Pan, Jiayaun Zhang

For the discrete-time switch–walk–switch walk on Z₂ wr Z, started with all lamps off and walker positions 0 or 2, the total-variation distance between the time-t laws is comparable to t^(−1/2), with constants independent of t.

AI involvementAI-autonomous

ImportanceModest

Technical verificationFormal (human)

Human understandingFew