Some Conditional Correlation Inequalities for Percolation and Related Processes
Abstract
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) randomcluster 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).
 Publication:

arXiv Mathematics eprints
 Pub Date:
 August 2004
 arXiv:
 arXiv:math/0408176
 Bibcode:
 2004math......8176V
 Keywords:

 Mathematics  Probability;
 Mathematics  Combinatorics;
 60C05;
 60K35;
 05C99