It is well-known in percolation theory (and intuitively plausible) that two events of the form ``there is an open path from $s$ to $a$' are positively correlated. We prove the (not intuitively obvious) fact that this is still true if we condition on an event of the form ``there is no open path from $s$ to $t$'.

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

van den Berg, J, & Kahn, J. (1999). A correlation inequality for connection events in percolation. CWI. Probability, Networks and Algorithms [PNA]. CWI.