The Annals of Applied Probability

A Triangle Inequality for Covariances of Binary FKG Random Variables

J. Van Den Berg and A. Gandolfi

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Abstract

For binary random variables $\sigma_1, \sigma_2, \ldots, \sigma_n$ that satisfy the well-known FKG condition, we show that the variances and covariances satisfy $\operatorname{Var}(\sigma_j) \operatorname{Cov}(\sigma_i, \sigma_k) \geq \operatorname{Cov}(\sigma_i, \sigma_j)\operatorname{Cov}(\sigma_j, \sigma_k),\quad 1 \leq i, j, k \leq n.$ This generalizes and improves a result by Graham for ferromagnetic Ising models with nonnegative external fields.

Article information

Source
Ann. Appl. Probab., Volume 5, Number 1 (1995), 322-326.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1177004843

Digital Object Identifier
doi:10.1214/aoap/1177004843

Mathematical Reviews number (MathSciNet)
MR1325056

Zentralblatt MATH identifier
0841.60011

JSTOR
links.jstor.org

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
FKG inequality correlation inequalities Ising model correlation length

Citation

Berg, J. Van Den; Gandolfi, A. A Triangle Inequality for Covariances of Binary FKG Random Variables. Ann. Appl. Probab. 5 (1995), no. 1, 322--326. doi:10.1214/aoap/1177004843. https://projecteuclid.org/euclid.aoap/1177004843


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