关于凯莱树上Potts模型的自由态要么是极值态要么是玻璃态
The free state for the Potts model on Cayley trees is either extremal or glassy
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中文总结 AI 辅助
研究凯莱树\(\mathbb{T}^d\)（\(d\geq2\)）上Potts模型的自由态，通过极限吉布斯测度与自由边界条件得到。证明自由态要么是极值态要么是玻璃态，还给出了伊辛模型自由态的相关推论。
中文摘要 AI 辅助
对于具有分支因子\(d\geq2\)的凯莱树\(\mathbb{T}^d\)上的Potts模型，我们考虑通过具有自由边界条件的极限吉布斯测度得到的自由态。我们证明自由态要么是极值态要么是玻璃态（即其分解为极值吉布斯测度包含不可数多个分量）。作为推论，\(\mathbb{T}^d\)上伊辛模型的自由态是玻璃态当且仅当逆温度\(\beta>\operatorname{arctanh}(1/\sqrt{d})\)；这将Gandolfo、Maes、Ruiz和Shlosman（2020）之前从非常低温度区域的结果推广到了整个自旋玻璃区域。
英文摘要
For the Potts model on the Cayley tree $\mathbb{T}^d$ with branching factor $d\geq 2$, we consider the free state which is obtained as the limiting Gibbs measure with free boundary conditions. We prove that the free state is either extremal or glassy (i.e., whose decomposition into extremal Gibbs measures contains uncountably many components). As a corollary, the free state for the Ising model on $\mathbb{T}^d$ is glassy if and only if the inverse temperature $β>\operatorname{arctanh}(1/\sqrt{d})$; this generalizes a previous result by Gandolfo, Maes, Ruiz and Shlosman (2020) from a very low temperature regime to the entire spin-glass regime.
