Chains with Complete Connections and One-Dimensional Gibbs Measures
Chains with Complete Connections and One-Dimensional Gibbs Measures
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具有完全连接和一维吉布斯测量的链
DOI：
10.1214/ejp.v9-149
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发表时间：
2003
影响因子：
1.4
通讯作者：
G. Maillard
中科院分区：
文献类型：
--
作者：
R. Fernández;G. Maillard
我们讨论一维吉布斯测度和离散时间过程（链）之间的关系。我们考虑有限字母（有限自旋）系统，可能带有语法（排除规则）。我们为随机过程建立条件来定义吉布斯测度，反之亦然。我们的条件概括了遍历马尔可夫链和域之间众所周知的等价结果，以及具有指数连续率的过程的已知吉布斯特征。我们的论点纯粹是概率性的；它们基于对条件概率（规范）的常规系统的研究。此外，我们讨论了链和域的唯一性准则的等价性，并为各自的有限体积条件概率系统的连续率建立了界限。作为辅助结果，我们证明了从单点调节开始的规范的（重新）构造定理，该定理适用于更一般的设置（一般自旋空间，规范不一定是吉布斯）。
We discuss the relationship between one-dimensional Gibbs measures and discrete-time processes (chains). We consider finite-alphabet (finite-spin) systems, possibly with a grammar (exclusion rule). We establish conditions for a stochastic process to define a Gibbs measure and vice versa. Our conditions generalize well known equivalence results between ergodic Markov chains and fields, as well as the known Gibbsian character of processes with exponential continuity rate. Our arguments are purely probabilistic; they are based on the study of regular systems of conditional probabilities (specifications). Furthermore, we discuss the equivalence of uniqueness criteria for chains and fields and we establish bounds for the continuity rates of the respective systems of finite-volume conditional probabilities. As an auxiliary result we prove a (re)construction theorem for specifications starting from single-site conditioning, which applies in a more general setting (general spin space, specifications not necessarily Gibbsian).