Rapid mixing for spin systems on graphs of girth at least five
Glauber dynamics for proper q-colourings mixes rapidly once q exceeds the maximum degree by any fixed factor, provided the graph has girth at least five.
- Lab
- Independent
- Model
- GPT-5.6 Sol Ultra
- Field
- Computer science
- Date
- 2026-08-26
- Human collaborators
- Xiaoyu Chen, Kuikui Liu
- Problem posed
- 1995 · open 31 yrs
Sources
Original work
Independent commentary
What was found
For every fixed positive delta, the Glauber dynamics on proper q-colourings is shown to mix rapidly when q is at least (1 + delta) times the maximum degree, the graph has girth at least five, and the maximum degree is large in terms of delta. The result extends to general multi-spin systems satisfying a local spectral contraction condition, including the anti-ferromagnetic Potts model. The engine is a new spectral local-to-global principle for multi-spin systems on graphs of girth at least five, together with a Fourier analysis for Glauber dynamics on a star.
Novelty check
The target is Jerrum's long-standing question about the colouring threshold, and the paper situates itself explicitly against the recent chain it builds on: the Bochner-Bakry-Emery proof of the spectral-gap criterion of Chen, Chen, Chen, Yin and Zhang, and the trickle-down and spectral independence literature it names. The girth-five restriction is stated as a hypothesis rather than removed, so the claim is a partial advance on a known frontier and is positioned as such. No prior result at this threshold under these hypotheses appears.
Caveats and known objections
A preprint days old at entry, unrefereed, not formalized, and with no independent check on record. The result is partial: girth at least five and a large maximum degree are genuine hypotheses, not technicalities. Autonomy is graded collaborative on the authors' own narrative, which describes several rounds of interaction rather than a single handoff: they first asked the model to redo known proofs by the Bochner identity, then brought it the open question, at which point it proposed an affirmative proof strategy; the human authors then verified the argument, generalized the proof with further assistance, and streamlined the paper.
Also recorded at
vibemathedrapid-mixing-for-spin-systems-on-graphs-of-girth-at-least-five
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
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Entry history (1 event)
- AddedEntered the registry graded Author verified and Collaborative.
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Cite this entry
whataifound.org. (2026). Rapid mixing for spin systems on graphs of girth at least five. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-26-spin-systems-girth-fiveBibTeX
@misc{whataifound-independent-2026-five,
title = {Rapid mixing for spin systems on graphs of girth at least five},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Author verified. Autonomy: Collaborative.},
url = {https://whataifound.org/finding/2026-08-26-spin-systems-girth-five}
}