论文标题
均质多组分Curie-Weiss-Potts模型的动力相变
Dynamical phase transition for the homogeneous multi-component Curie-Weiss-Potts model
论文作者
论文摘要
在本文中，我们研究了具有$ q \ geq 3 $旋转的均质多组分居里 - 韦斯 - 彼得。该模型是在完整的图$ k_ {nm} $上定义的，其顶点集均等分为$ n $的$ m $组件。对于配置，$σ：\ {1，\ cdots，nm \} \ to \ {1，\ cdots，q \}，$ gibbs Memues由$$定义 μ_{n，β}（σ）= \ frac {1} {z_ {z_ {n，β}} \ exp \ big（\fracβ{n} \ sum_ {v，v，w = 1} \ Mathbb {1} _ {\ {σ（v）=σ（w）\}} \ big），$ $，其中$ z_ {n，β} $是一个归一化的常数，$β> 0 $是相反的温度参数。相互作用系数为$ \ MATHCAL {J}（v，w）= \ frac {j} {1 +（m-1）λ} $，对于同一组件中的$ v，w $，$ v，$ \ mathcal {j}（j}（v，v，v，w，w，w）= \ frac {jλ} $} $λ\ in（0，1）$是组件间相互作用与组件内交互的相对强度，而$ j> 0 $是有效的相互作用强度。我们确定在关键反向温度$β_{\ operatoRatorName {cr}} =β_{s}（q）/j $的动态相变，其中$β_{s}（q）$是最大的逆温度，可保证在curie-weiss-weiss-weiss-weiss-weiss-weiss-weiss-weiss-weiss-weisiss-weys-weys-weys-weys-weys-weys-weys-weys-weys-weys-weys-weys-weys-weyspotts model arxiv arxiv arxiv：12004.4403。通过将汇总路径方法ARXIV：1312.6728扩展到我们的多组分设置，我们证明了在高温方案中的混合时间$ o（n \ log n）$混合时间在低温级别的低温状态下，低召唤状态$β>β>β>β>β>β> s}（q）/$ simistions/j nime nime nime nime nime nime nime nime nime nime nime nime nime nime。这是多组分Potts模型中动态相变的第一个结果。
In this paper, we study the homogeneous multi-component Curie-Weiss-Potts model with $q \geq 3$ spins. The model is defined on the complete graph $K_{Nm}$, whose vertex set is equally partitioned into $m$ components of size $N$. For a configuration $σ: \{1, \cdots, Nm\} \to \{1, \cdots, q\},$ the Gibbs measure is defined by $$ μ_{N,β}(σ) =\frac{1}{Z_{N,β}} \exp\Big(\fracβ{N} \sum_{v,w=1}^{Nm}\mathcal{J}(v,w)\, \mathbb{1}_{\{σ(v)=σ(w)\}}\Big), $$ where $Z_{N, β}$ is a normalizing constant, and $β>0$ is the inverse temperature parameter. The interaction coefficients are $ \mathcal{J}(v, w) = \frac{J}{1 + (m-1) λ}$, for $v, w$ in the same component, and $\mathcal{J}(v, w) = \frac{J λ}{1 + (m-1)λ}$ for $v, w$ in the different components, where $λ\in (0, 1)$ is the relative strength of inter-component interaction to intra-component interaction, and $J>0$ is the effective interaction strength. We identify a dynamical phase transition at the critical inverse temperature $β_{\operatorname{cr}} = β_{s}(q)/J$, where $β_{s}(q)$ is maximal inverse temperature guaranteeing a unique critical point of the free energy in the Curie-Weiss-Potts model arXiv:1204.4503. By extending the aggregate path method arXiv:1312.6728 to our multi-component setting, we prove $O(N \log N)$ mixing time in the high-temperature regime $β<β_{s}(q)/J.$ In the low-temperature regime $β> β_{s}(q)/J,$ we further show exponential mixing time by a metastability. This is the first result for the dynamical phase transition in the multi-component Potts model.
