Domination by product measures

• 成果类型: Article
• 署名作者: Liggett, TM; Schonmann, RH; Stacey, AM
• 署名单位: University of California System; University of California Los Angeles
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 71-95
• 关键词: large deviations; percolation

摘要

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.