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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
##### Keywords:
approximate zero-one law; threshold effect; product measure
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