TY - JOUR

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

AU - Kahn, Jeff

AU - Kayll, P. Mark

N1 - Funding Information:
Mathematics Subject Classification (1991): 05C70, 05C65, 60C05; 52B12, 82B20. * Supported in part by NSF. t This work forms part of the author's doctoral dissertation \[16\];s ee also \[17\]. The author gratefully acknowledges NSERC for partial support in the form of a 1967 Science and Engineering Scholarship.

PY - 1997

Y1 - 1997

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

U2 - 10.1007/BF01215919

DO - 10.1007/BF01215919

M3 - Article

AN - SCOPUS:0031439717

SN - 0209-9683

VL - 17

SP - 369

EP - 391

JO - Combinatorica

JF - Combinatorica

IS - 3

ER -