Some conditional correlation inequalities for percolation and related processes

Abstract

AbstractConsider ordinary bond percolation on a finite or countably infinite graph. Let s, t, a, and b be vertices. An earlier paper (J. Van den Berg and J. Kahn, Ann Probab 29 (2001), 123–126) 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 et al. (Stoch Proc Appl 67 (1997), 213–225). © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006