The Annals of Probability

On Russo's Approximate Zero-One Law

Michel Talagrand

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Abstract

Consider the product measure $\mu_p$ on $\{0, 1\}^n$, when 0 $(\operatorname{resp}. 1)$ is given weight $1 - p (\operatorname{resp}. p)$. Consider a monotone subset $A$ of $\{0, 1\}^n$. We give a precise quantitative form to the following statement: if $A$ does not depend much on any given coordinate, $d\mu_p(A)/dp$ is large. Thus, in that case, there is a threshold effect and $\mu_p(A)$ jumps from near 0 to near 1 in a small interval.

Article information

Source
Ann. Probab. Volume 22, Number 3 (1994), 1576-1587.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988612

Digital Object Identifier
doi:10.1214/aop/1176988612

Mathematical Reviews number (MathSciNet)
MR1303654

Zentralblatt MATH identifier
0819.28002

JSTOR
links.jstor.org

Subjects
Primary: 28A35: Measures and integrals in product spaces
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Approximate zero-one law threshold effect influence of variables

Citation

Talagrand, Michel. On Russo's Approximate Zero-One Law. Ann. Probab. 22 (1994), no. 3, 1576--1587. doi:10.1214/aop/1176988612. https://projecteuclid.org/euclid.aop/1176988612.


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