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PreprintCC BY 4.0arXiv

A Spectral Local-to-Global Principle for Spin Systems on Graphs with Girth At Least Five

  • Xiaoyu Chen
  • Kuikui Liu
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Abstract

It is proved that, for every δ∈ (0,1), the Glauber dynamics for the uniform distribution on proper q-colorings is rapidly mixing when q ≥ (1+δ)Δ and the underlying graph has girth at least $5$ and maximum degree Δ= Ωδ(1). This result also extends to general multi-spin systems satisfying a local spectral contraction condition, including the anti-ferromagnetic Potts model with q≥ (1+δ)(1−β)Δ. These results are achieved by a new spectral local-to-global principle on graphs with girth at least five for general multi-spin systems, and a novel Fourier analysis for Glauber dynamics on a star. The main ideas behind all the proofs were developed through several rounds of interaction with GPT-5.6 Sol Ultra.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026spectral,
  title = {{A Spectral Local-to-Global Principle for Spin Systems on Graphs with Girth At Least Five}},
  author = {Xiaoyu Chen and Kuikui Liu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2608.25491},
  eprint = {2608.25491},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.25491},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Spectral Local-to-Global Principle for Spin Systems on Graphs with Girth At Least Five
AU  - Xiaoyu Chen
AU  - Kuikui Liu
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2608.25491
UR  - https://arxiv.org/abs/2608.25491
AB  - It is proved that, for every δ∈ (0,1), the Glauber dynamics for the uniform distribution on proper q-colorings is rapidly mixing when q ≥ (1+δ)Δ and the underlying graph has girth at least $5$ and maximum degree Δ= Ω_δ(1). This result also extends to general multi-spin systems satisfying a local spectral contraction condition, including the anti-ferromagnetic Potts model with q≥ (1+δ)(1−β)Δ. These results are achieved by a new spectral local-to-global principle on graphs with girth at least five for general multi-spin systems, and a novel Fourier analysis for Glauber dynamics on a star. The main ideas behind all the proofs were developed through several rounds of interaction with GPT-5.6 Sol Ultra.
ER  -