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)

Interacting random processes; statistical mechanics type models; percolation theory (msc 60K35), None of the above, but in MSC2010 section 05Cxx (msc 05C99), Combinatorial probability (msc 60C05)
Logistics (theme 3), Energy (theme 4)
CWI. Probability, Networks and Algorithms [PNA]

van den Berg, J, Häggström, O, & Kahn, J. (2004). Some conditional correlation inequalities for percolation and related processes. CWI. Probability, Networks and Algorithms [PNA]. CWI.