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Some Conditional Correlation Inequalities for Percolation and Related Processes

Jacob van den Berg·Olle Haggstrom·Jeff Kahn
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Abstract

Consider ordinary bond percolation on a finite or countably infinite graph. Let s, t, a and b be vertices. An earlier paper proved the (nonintuitive) result that, conditioned on the event that there is no open path from s to t, the two events "there is an open path from s to a" and "there is an open path from s to b" are positively correlated. In the present paper we further investigate and generalize the theorem of which this result was a consequence. This leads to results saying, informally, that, with the above conditioning, the open cluster of s is conditionally positively (self-)associated and that it is conditionally negatively correlated with the open cluster of t. We also present analogues of some of our results for (a) random-cluster measures, and (b) directed percolation and contact processes, and observe that the latter lead to improvements of some of the results in a paper of Belitsky, Ferrari, Konno and Liggett (1997).

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Jacob van den Berg, Olle Haggstrom, Jeff Kahn. 2004-08-13. Some Conditional Correlation Inequalities for Percolation and Related Processes. https://arxiv.org/abs/math/0408176

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