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 $t$'.

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CWI
CWI. Probability, Networks and Algorithms [PNA]
Stochastics

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