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On Russo’s approximate zero-one law. (English) Zbl 0819.28002
A subset \(A\) of \(\{0, 1\}^ n\) is called monotone if \(x_ i\leq y_ i\), \(i=1, \dots, n\), for \((x_ 1,\dots, x_ n)\in A\) and some \((y_ 1,\dots, y_ n)\in \{0,1 \}^ n\) implies \((y_ 1, \dots, y_ n)\in A\). Furthermore, the set \(A_ i\) consists of all \((x_ 1, \dots, x_ n)\in A\) such that \((x_ 1,\dots, x_{i-1}, 1-x_ i, x_{i+1}, \dots, x_ n)\not\in A\) is satisfied, \(1\leq i\leq n\), and \(\mu_ p\) stands for the product measure defined by \(\mu_ p (\{( x_ 1,\dots, x_ n)\})= p^ k (1-p )^{n-k}\), \((x_ 1,\dots, x_ n)\in \{0,1 \}^ n\), \(k= \text{card} \{i\in \{1,\dots, n\}\): \(x_ i=1\}\), \(p\in (0,1)\). The following inequality, which is sharp for any \(p\in (0, 1)\), is the main result: \[ \mu_ p (A) (1- \mu_ p (A))\leq K(1-p) \log {2\over {p(1- p)}} \sum_{i=1}^ n {{\mu_ p (A_ i)} \over {\log ([ (1-p) \mu_ p (A_ i) ]^{-1} )}} \] is vaid for all monotone subsets \(A\) of \(\{0, 1\}^ n\) and any \(p\in (0,1)\), where \(K\) is some universal constant.

MSC:
28A35 Measures and integrals in product spaces
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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