关于一般一维XY模型：正温度与零温度，选择与非选择
On the general one-dimensional XY Model: positive and zero temperature, selection and non-selection
摘要：
我们认为$(M,d)$是一个连通的紧致流形，用$\mathcal{B}_i$表示由$$x=(... x_{-3},x_{-2},x_{-1},x_0,x_1,x_2,x_3,...),$$表示的序列的伯努利空间$M^{\Z}$，其中$x_i$属于空间（字母表）$M$ .这里特别关注单位圆$M=\mathbb{S}^1$的情况。在一维格点$\mathbb{N}$的类似问题也被认为是。在这种情况下，我们考虑势$A: {\cal B}=M^\mathbb{N} \to \mathbb{R}.$。令$A: \mathcal{B}_i \rar \R$是在伯努利空间$\mathcal{B}_i$中定义的{ \it可观测}或{ \it势}。潜在的$A$描述了在一维晶格$M^\mathbb{Z}$网站之间的相互作用。给定一个温度$T$，我们分析了在${\cal B}_i$上的一个确定的概率测度--吉布斯态$\hatμ_{\frac{1}{T} A}$的主要性质.我们将这种设置称为“通用XY模型”。为了进行分析，我们考虑与$\frac{1}{T} A$相关联的Ruelle算子，并且，我们在这个过程中得到主本征函数$ψ_{\frac{1}{T} A}$。后来，我们分析选择问题时，温度趋于零：a）存在，或不存在，极限（一致收敛）$$V:=\lim_{T\to 0} T\, \log(ψ_{\frac{1}{T} A}),\,\,\,\,\text{a question about selection of subaction},$$和，b）存在，或不存在，极限（弱$^*$意义上）$$\tildeμ:=\lim_{T\to 0} \hatμ_{\frac{1}{T}\, A},\,\,\,\,\text{a question about selection of measure}.$$的存在性子行动和遍历优化的其他属性也被认为是。
英文摘要：
We consider $(M,d)$ a connected and compact manifold and we denote by $\mathcal{B}_i$ the Bernoulli space $M^{\Z}$ of sequences represented by $$x=(... x_{-3},x_{-2},x_{-1},x_0,x_1,x_2,x_3,...),$$ where $x_i$ belongs to the space (alphabet) $M$. The case where $M=\mathbb{S}^1$, the unit circle, is of particular interest here. The analogous problem in the one-dimensional lattice $\mathbb{N}$ is also considered. %In this case we consider the potential $A: {\cal B}=M^\mathbb{N} \to \mathbb{R}.$ Let $A: \mathcal{B}_i \rar \R$ be an {\it observable} or {\it potential} defined in the Bernoulli space $\mathcal{B}_i$. The potential $A$ describes an interaction between sites in the one-dimensional lattice $M^\mathbb{Z}$. Given a temperature $T$, we analyze the main properties of the Gibbs state $\hatμ_{\frac{1}{T} A}$ which is a certain probability measure over ${\cal B}_i$. We denote this setting "the general XY model". In order to do our analysis we consider the Ruelle operator associated to $\frac{1}{T} A$, and, we get in this procedure the main eigenfunction $ψ_{\frac{1}{T} A}$. Later, we analyze selection problems when temperature goes to zero: a) existence, or not, of the limit (on the uniform convergence) $$V:=\lim_{T\to 0} T\, \log(ψ_{\frac{1}{T} A}),\,\,\,\,\text{a question about selection of subaction},$$ and, b) existence, or not, of the limit (on the weak$^*$ sense) $$\tildeμ:=\lim_{T\to 0} \hatμ_{\frac{1}{T}\, A},\,\,\,\,\text{a question about selection of measure}.$$ The existence of subactions and other properties of Ergodic Optimization are also considered.
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