广义划分和颜色模型
Generalized Divide and Color models
摘要：
本文首先对“广义分色模型”进行了研究。一个非常特别有趣的例子是Olle Häggström介绍和研究的“划分和颜色模型”（这是我们使用的名称的动机）。在这个广义模型中，我们从一个有限的或可数的集合$V$，$V$的一个随机划分和一个参数$p\in [0,1]$开始。相应的广义划分和颜色模型是由$V$索引的$\{0,1\}$值过程，通过独立地，对于随机划分中选择的每个划分元素，以概率$p$，将值1分配给划分元素的所有元素，并且以概率$1-p$，将值0分配给划分元素的所有元素而获得。我们在这里研究的一些问题如下。在什么情况下不同的随机分区可以产生相同的颜色过程？关于可交换的随机划分，我们能说些什么呢？一个色彩过程随机控制的积测度集是什么？对于具有平移不变的随机划分，其颜色过程具有什么样的遍历性质？研究这些过程的动机是双重的;一方面，我们相信这是一类非常自然和有趣的过程，值得研究，另一方面，许多变化很大的、经过充分研究的过程实际上属于这一类，例如（1）伊辛模型，（2）模糊波茨模型，（3）选举模型的平稳分布，（4）随机场景中的随机游动，当然还有（5）原始的分割和颜色模型。
英文摘要：
In this paper, we initiate the study of "Generalized Divide and Color Models". A very special interesting case of this is the "Divide and Color Model" (which motivates the name we use) introduced and studied by Olle Häggström. In this generalized model, one starts with a finite or countable set $V$, a random partition of $V$ and a parameter $p\in [0,1]$. The corresponding Generalized Divide and Color Model is the $\{0,1\}$-valued process indexed by $V$ obtained by independently, for each partition element in the random partition chosen, with probability $p$, assigning all the elements of the partition element the value 1, and with probability $1-p$, assigning all the elements of the partition element the value 0. Some of the questions which we study here are the following. Under what situations can different random partitions give rise to the same color process? What can one say concerning exchangeable random partitions? What is the set of product measures that a color process stochastically dominates? For random partitions which are translation invariant, what ergodic properties do the resulting color processes have? The motivation for studying these processes is twofold; on the one hand, we believe that this is a very natural and interesting class of processes that deserves investigation and on the other hand, a number of quite varied well-studied processes actually fall into this class such as (1) the Ising model, (2) the fuzzy Potts model, (3) the stationary distributions for the Voter Model, (4) random walk in random scenery and of course (5) the original Divide and Color Model.
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