Cutoff for Kac's Walk on the Sphere
The discrete-time Kac walk on started from a coordinate vector exhibits total variation cutoff at , where is set by the speed of the leftmost particle in a branching random walk. The cutoff is therefore not at the conjectured .
- Result
- Proved(see note)
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Argument
- Field
- Markov chains
- Posed by
- Mark Kac
- Year posed
- 1956
- Years open
- 70y
- Solved
- 2026-07-15
- Model
- ChatGPT
- Vendor
- OpenAI
- Collaborators
- Vishesh Jain, Clayton Mizgerd
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 20 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
the conjectured cutoff location of 2n log n is wrong
What the AI did
The acknowledgement says the authors used ChatGPT extensively, at the level of a co-author, for brainstorming, help with proofs, literature review, checking for mistakes, writing code, and preparing the manuscript. It does not separate which parts came from where.
Verification
arXiv preprint; not yet peer-reviewed.
Source
arXiv:2607.13401 - Total variation cutoff for Kac's walk on the sphere