Mixing Time of Kac's Walk on the Rotation Group
Kac's walk on the rotation group, introduced by Hastings in 1970, is a central high-dimensional Markov chain in statistical physics and computational science. The paper proves it mixes in steps, the conjectured optimal rate, closing the gap left by a long line of successive improvements.
- Result
- Proved
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Argument
- Field
- Markov chain mixing
- Posed by
- Walk introduced by W. K. Hastings; optimal rate the standing target of the mixing-time literature
- Year posed
- 1970
- Years open
- 56y
- Solved
- 2026-04-26
- Model
- GPT Pro 5.4
- Vendor
- OpenAI
- Collaborators
- Natesh S. Pillai, Aaron Smith
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 20 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
"Much of this paper was developed in collaboration with GPT Pro 5.4. In particular, the idea of using matrix martingale inequalities" came from that collaboration. Pillai and Smith are the authors of the previous best bounds for this walk.