Domination by product measures
Domination by product measures
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DOI：
10.1214/aop/1024404279
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发表时间：
1997
影响因子：
2.3
通讯作者：
T. Liggett;R. Schonmann;A. Stacey
中科院分区：
文献类型：
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作者：
T. Liggett;R. Schonmann;A. Stacey
研究了由有界度可数图的顶点所标的（0，1）值随机变量族。首先，我们证明，如果这些随机变量满足的属性，条件是发生在每个给定的网站的邻域外，在这个网站上看到1的概率至少是一个值p是足够大的，那么这个随机场主导的产品测量与正密度。此外，这个支配积测度的密度可以任意接近1，只要p足够接近1。接下来，我们解决的问题，获得临界值的p，定义为阈值以上的正密度积措施的统治是保证。对于顶点为整数且顶点间的边间隔不超过k个单位的图，这个临界值为1-kk/（k +1）k +1，并且存在一个不连续的转移.对于{0，1} Z上的其他类概率测度，也发现了类似的p的临界值.对于k-相依测度类，临界值也是1-k k/（k +1）k +1，有一个不连续的过渡。对于类的两个区组因子的临界值被证明是1/2和连续的过渡被证明在这种情况下发生。因此，在两区组因子和1依赖的情况下，临界值和过渡的性质是不同的。
We consider families of(0,1)-valued random variables indexed by the vertices of countable graphs with bounded degree. First we show that if these random variables satisfy the property that conditioned on what happens outside of the neighborhood of each given site, the probability of seeing a 1 at this site is at least a value p which is large enough, then this random field dominates a product measure with positive density. Moreover the density of this dominated product measure can be made arbitrarily close to 1, provided that p is close enough to 1. Next we address the issue of obtaining the critical value of p, defined as the threshold above which the domination by positive-density product measures is assured. For the graphs which have as vertices the integers and edges connecting vertices which are separated by no more than k units, this critical value is shown to be 1 - k k /(k + 1) k+1 , and a discontinuous transition is shown to occur. Similar critical values of p are found for other classes of probability measures on {0, 1} Z . For the class of k-dependent measures the critical value is again 1 - k k /(k + 1) k+1 , with a discontinuous transition. For the class of two-block factors the critical value is shown to be 1/2 and a continuous transition is shown to take place in this case. Thus both the critical value and the nature of the transition are different in the two-block factor and 1-dependent cases.