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arXiv · 2608.25491

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 $\delta \in (0,1)$, the Glauber dynamics for the uniform distribution on proper $q$-colorings is rapidly mixing when $q \geq (1+\delta)\Delta$ and the underlying graph has girth at least $5$ and maximum degree $\Delta = \Omega_\delta(1)$. This result also extends to general multi-spin systems satisfying a $\textit{local spectral contraction}$ condition, including the anti-ferromagnetic Potts model with $q\geq (1+\delta)(1-\beta)\Delta$. 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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Xiaoyu Chen, Kuikui Liu. 2026-08-26. A Spectral Local-to-Global Principle for Spin Systems on Graphs with Girth At Least Five. https://arxiv.org/abs/2608.25491

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