A triangle inequality for covariances of binary FKG random variables. (English) Zbl 0841.60011

van den Berg, J.; Gandolfi, A.

Ann. Appl. Probab. 5, No. 1, 322-326 (1995).

For binary random variables \(\sigma_1, \dots, \sigma_n\) whose joint distribution is denoted by \(\mu\), the FKG condition reads \(\mu(\alpha \vee \beta) \mu(\alpha \wedge \beta) \geq \mu(\alpha)\mu(\beta)\) where the \(i\)-th coordinate of the configuration \((\alpha \vee \beta)\) is the minimum of the \(i\)-th coordinates of \(\alpha\) and \(\beta\) and similarly the \(i\)-th coordinate of the configuration \((\alpha \wedge \beta)\) is the maximum of the \(i\)-th coordinates of \(\alpha\) and \(\beta\). The authors show that for random variables satisfying the FKG condition the following inequality holds \[ \text{Var}(\sigma_j) \text{Cov}(\sigma_i, \sigma_k) \geq \text{Cov}(\sigma_i, \sigma_j) \text{Cov}(\sigma_j,\sigma_k). \]

Reviewer: P.A.Ferrari (São Paulo)

MSC:

60E15	Inequalities; stochastic orderings
60K35	Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

FKG inequality; correlation inequalities; Ising model; correlation length