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