## The Annals of Probability

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

### On Russo's Approximate Zero-One Law

#### 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**

http://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. http://projecteuclid.org/euclid.aop/1176988612.