ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

13. Combinatorics

1991 MS Classification: 05, 60, 52, 82 Kayll, P. Mark, University of Montana, USA: Stochastic independence properties of hard-core distributions A probability measure p on the set M of matchings in a multigraph Γ is hard-core if for some λ : Γ → [0, ∞), the probability p(M) of M ∈ M is proportional to ∏_{A∈M} λ(A). Such distributions enjoy substantial approximate stochastic independence properties. For example, if M is chosen according to the hard-core distribution p, MP(Γ) is the matching polytope of Γ, δ > 0 is fixed, and if the vector of marginals, (Pr(A ∈ M) : A an edge of Γ), is in (1 − δ)MP(Γ), then \frac{Pr(A,B∈M)}{Pr(A∈M) Pr(B∈M)} → 1 as the distance between A, B ∈ Γ tends to infinity. Applications of these results include the resolutions of several questions involving asymptotic colouring behaviour of multigraphs. The talk is based on joint work with Jeff Kahn. Reference: J. Kahn and P. M. Kayll, On the stochastic independence properties of hard-core distributions, Combinatorica 17 (1997), 369–391.