Rapid mixing for spin systems on graphs of girth at least five

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This event attributed to Kuikui Liu

This event attributed to Xiaoyu Chen

This event attributed to GPT-5.6 Sol Ultra

Rapid mixing for spin systems on graphs of girth at least five parent of this event

Rapid mixing for spin systems on graphs of girth at least five evidence for this event

For fixed $\delta\in(0,1)$ and all sufficiently large $\Delta$ depending only on $\delta$, Glauber dynamics for proper $q$-colorings mixes rapidly on every graph of girth at least $5$ whenever $q\ge(1+\delta)\Delta$: spectral gap $\Omega_\delta(1/n)$ and $t_{\mathrm{mix}}(\varepsilon)=O_\delta(n^2\log q+n\log(1/\varepsilon))$. An analogous theorem holds for the anti-ferromagnetic Potts model at $q\ge(1+\delta)(1-\beta)\Delta$. What it does and does not improve. On girth it is a large gain: previous results near the $(1+\delta)\Delta$ threshold needed girth at least eleven (Hayes-Vigoda, extended to constant degrees by Jain-Mizgerd-Vigoda). On the mixing rate it is weaker - those give optimal $O(n\log n)$, this gives $O(n^2\log q)$. It does not touch the folklore conjecture that mixing is rapid on every graph for $q\ge\Delta+2$; Remark 7 names spanning 4-cycles as the obstruction. parent of this event

VibeMathed record: Rapid mixing for spin systems on graphs of girth at least five evidence for this event