### Abstract

A probability measure p on the set M of matchings in a graph (or, more generally 2-bounded hypergraph) Γ is hard-core if for some λ:Γ → [0, ∞), the probability p(M) of M ∈ M is proportional to Π_{A∈M} λ(A). We show that such distributions enjoy substantial approximate stochastic independence properties. This is based on showing that, with M chosen according to the hard-core distribution p, MP (Γ) the matching polytope of Γ, and δ > 0, if the vector of marginals, (Pr(A ∈ M): A an edge of Γ), is in (1 - δ)MP (Γ), then the weights λ(A) are bounded by some A(δ). This eventually implies, for example, that under the same assumption, with δ fixed, Pr(A,B∈M)/Pr(A∈M)Pr(B∈M) → 1 as the distance between A, B ∈ Γ tends to infinity. Thought to be of independent interest, our results have already been applied in the resolutions of several questions involving asymptotic behaviour of graphs and hypergraphs (see [14, 16], [11]-[13]).

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

Pages (from-to) | 369-391 |

Number of pages | 23 |

Journal | Combinatorica |

Volume | 17 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 1997 |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Computational Mathematics

### Cite this

*Combinatorica*,

*17*(3), 369-391. https://doi.org/10.1007/BF01215919

}

*Combinatorica*, vol. 17, no. 3, pp. 369-391. https://doi.org/10.1007/BF01215919

**On the stochastic independence properties of hard-core distributions.** / Kahn, Jeffry; Kayll, P. Mark.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the stochastic independence properties of hard-core distributions

AU - Kahn, Jeffry

AU - Kayll, P. Mark

PY - 1997/1/1

Y1 - 1997/1/1

N2 - A probability measure p on the set M of matchings in a graph (or, more generally 2-bounded hypergraph) Γ is hard-core if for some λ:Γ → [0, ∞), the probability p(M) of M ∈ M is proportional to ΠA∈M λ(A). We show that such distributions enjoy substantial approximate stochastic independence properties. This is based on showing that, with M chosen according to the hard-core distribution p, MP (Γ) the matching polytope of Γ, and δ > 0, if the vector of marginals, (Pr(A ∈ M): A an edge of Γ), is in (1 - δ)MP (Γ), then the weights λ(A) are bounded by some A(δ). This eventually implies, for example, that under the same assumption, with δ fixed, Pr(A,B∈M)/Pr(A∈M)Pr(B∈M) → 1 as the distance between A, B ∈ Γ tends to infinity. Thought to be of independent interest, our results have already been applied in the resolutions of several questions involving asymptotic behaviour of graphs and hypergraphs (see [14, 16], [11]-[13]).

AB - A probability measure p on the set M of matchings in a graph (or, more generally 2-bounded hypergraph) Γ is hard-core if for some λ:Γ → [0, ∞), the probability p(M) of M ∈ M is proportional to ΠA∈M λ(A). We show that such distributions enjoy substantial approximate stochastic independence properties. This is based on showing that, with M chosen according to the hard-core distribution p, MP (Γ) the matching polytope of Γ, and δ > 0, if the vector of marginals, (Pr(A ∈ M): A an edge of Γ), is in (1 - δ)MP (Γ), then the weights λ(A) are bounded by some A(δ). This eventually implies, for example, that under the same assumption, with δ fixed, Pr(A,B∈M)/Pr(A∈M)Pr(B∈M) → 1 as the distance between A, B ∈ Γ tends to infinity. Thought to be of independent interest, our results have already been applied in the resolutions of several questions involving asymptotic behaviour of graphs and hypergraphs (see [14, 16], [11]-[13]).

UR - http://www.scopus.com/inward/record.url?scp=0031439717&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031439717&partnerID=8YFLogxK

U2 - 10.1007/BF01215919

DO - 10.1007/BF01215919

M3 - Article

VL - 17

SP - 369

EP - 391

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 3

ER -