Some conditional correlation inequalities for percolation and related processes

Consider 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)