### Abstract

Recently, van den Berg and Jonasson gave the first substantial extension of the BK inequality for non-product measures: they proved that, for k-out-of-n measures, the probability that two increasing events occur disjointly is at most the product of the two individual probabilities. We show several other extensions and modifications of the BK inequality. In particular, we prove that the antiferromagnetic Ising Curie-Weiss model satisfies the BK inequality for all increasing events. We prove that this also holds for the Curie-Weiss model with three-body interactions under the so-called negative lattice condition. For the ferromagnetic Ising model we show that the probability that two events occur 'cluster-disjointly' is at most the product of the two individual probabilities, and we give a more abstract form of this result for arbitrary Gibbs measures. The above cases are derived from a general abstract theorem whose proof is based on an extension of the Fortuin-Kasteleyn random-cluster representation for all probability distributions and on a 'folding procedure' which generalizes an argument of Reimer.

Original language | English (US) |
---|---|

Pages (from-to) | 157-181 |

Number of pages | 25 |

Journal | Probability Theory and Related Fields |

Volume | 157 |

Issue number | 1-2 |

DOIs | |

State | Published - Oct 1 2013 |

### Fingerprint

### Keywords

- Arm events
- BK inequality
- Curie-Weiss model
- Foldings
- Negative dependence
- Random-cluster representation Gibbs distribution

### ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Probability Theory and Related Fields*,

*157*(1-2), 157-181. https://doi.org/10.1007/s00440-012-0452-1

**BK-type inequalities and generalized random-cluster representations.** / van den Berg, J.; Gandolfi, Alberto.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 157, no. 1-2, pp. 157-181. https://doi.org/10.1007/s00440-012-0452-1

}

TY - JOUR

T1 - BK-type inequalities and generalized random-cluster representations

AU - van den Berg, J.

AU - Gandolfi, Alberto

PY - 2013/10/1

Y1 - 2013/10/1

N2 - Recently, van den Berg and Jonasson gave the first substantial extension of the BK inequality for non-product measures: they proved that, for k-out-of-n measures, the probability that two increasing events occur disjointly is at most the product of the two individual probabilities. We show several other extensions and modifications of the BK inequality. In particular, we prove that the antiferromagnetic Ising Curie-Weiss model satisfies the BK inequality for all increasing events. We prove that this also holds for the Curie-Weiss model with three-body interactions under the so-called negative lattice condition. For the ferromagnetic Ising model we show that the probability that two events occur 'cluster-disjointly' is at most the product of the two individual probabilities, and we give a more abstract form of this result for arbitrary Gibbs measures. The above cases are derived from a general abstract theorem whose proof is based on an extension of the Fortuin-Kasteleyn random-cluster representation for all probability distributions and on a 'folding procedure' which generalizes an argument of Reimer.

AB - Recently, van den Berg and Jonasson gave the first substantial extension of the BK inequality for non-product measures: they proved that, for k-out-of-n measures, the probability that two increasing events occur disjointly is at most the product of the two individual probabilities. We show several other extensions and modifications of the BK inequality. In particular, we prove that the antiferromagnetic Ising Curie-Weiss model satisfies the BK inequality for all increasing events. We prove that this also holds for the Curie-Weiss model with three-body interactions under the so-called negative lattice condition. For the ferromagnetic Ising model we show that the probability that two events occur 'cluster-disjointly' is at most the product of the two individual probabilities, and we give a more abstract form of this result for arbitrary Gibbs measures. The above cases are derived from a general abstract theorem whose proof is based on an extension of the Fortuin-Kasteleyn random-cluster representation for all probability distributions and on a 'folding procedure' which generalizes an argument of Reimer.

KW - Arm events

KW - BK inequality

KW - Curie-Weiss model

KW - Foldings

KW - Negative dependence

KW - Random-cluster representation Gibbs distribution

UR - http://www.scopus.com/inward/record.url?scp=84884206509&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884206509&partnerID=8YFLogxK

U2 - 10.1007/s00440-012-0452-1

DO - 10.1007/s00440-012-0452-1

M3 - Article

VL - 157

SP - 157

EP - 181

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 1-2

ER -