Proof of the van den Berg-Kesten conjecture. (English) Zbl 0947.60093

Reimer, David

Comb. Probab. Comput. 9, No. 1, 27-32 (2000).

Summary: We prove the following conjecture of J. van den Berg and H. Kesten [J. Appl. Probab. 22, 556-569 (1985; Zbl 0571.60019)]. For any events \({\mathcal A}\) and \({\mathcal B}\) in a product probability space, \(\text{Prob}({\mathcal A}\square{\mathcal B})\leq \text{Prob}({\mathcal A}) \text{Prob}({\mathcal B})\), where \({\mathcal A} \square {\mathcal B}\) is the event that \({\mathcal A}\) and \({\mathcal B}\) occur ‘disjointly’.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60E15	Inequalities; stochastic orderings