On the stochastic independence properties of hard-core distributions

Abstract

A probability measurep on the set μ of matchings in a graph (or, more generally 2-bounded hypergraph) Γ ishard-core if for some λ: Γ→[0,∞), the probabilityp(M) ofM∈μ is proportional to\(\prod\nolimits_{A_ \in M} {\lambda (A)}\). We show that such distributions enjoy substantial approximate stochastic independence properties. This is based on showing that, withM chosen according to the hard-core distributionp, MP (Γ) the matching polytope of Γ, and σ>0, if the vector ofmarginals, (Pr(AM):A an edge of Γ), is in (1−σ) MP (Γ), then the weights λ(A) are bounded by someA(σ). This eventually implies, for example, that under the same assumption, with σ fixed,\(\frac{{\Pr (A,B \in M)}}{{\Pr (A \in M)\Pr (B \in M)}} \to 1\) as the distance betweenA, 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]).

This is a preview of subscription content, access via your institution.

References

  1. [1]

    N. Alon andJ. Spencer:The Probabilistic Method, Wiley, New York, 1992.

    Google Scholar 

  2. [2]

    J. van den Berg andJ. E. Steif: Percolation and the hard-core lattice gas model,Stochastic Process. Appl.,49 (1994), 179–197.

    Google Scholar 

  3. [3]

    C. Berge:Hypergraphs: Combinatorics of Finite Sets, North-Holland, Amsterdam, 1989.

    Google Scholar 

  4. [4]

    B. Bollobás:Extremal Graph Theory, Academic Press, London, 1978.

    Google Scholar 

  5. [5]

    J. Edmonds: Maximum matching and a polyhedron with 0,1-vertices,J. Res. Nat. Bur. Standards (B),69 (1965), 125–130.

    Google Scholar 

  6. [6]

    Z. Füredi: Matchings and covers in hypergraphs,Graphs Combin.,4 (1988), 115–206.

    Google Scholar 

  7. [7]

    C. D. Godsil andI. Gutman: On the theory of the matching polynomial,J. Graph Theory,5 (1981), 137–144.

    Google Scholar 

  8. [8]

    C. D. Godsil:Algebraic Combinatorics, Chapman and Hall, New York, 1993.

    Google Scholar 

  9. [9]

    O. J. Heilmann andE. H. Lieb: Monomers and dimers,Phys. Rev. Letters,24 (1970), 1412–1414.

    Google Scholar 

  10. [10]

    O. J. Heilmann andE. H. Lieb: Theory of monomer-dimer systems,Commun. Math. Phys.,25 (1972), 190–232.

    Google Scholar 

  11. [11]

    J. Kahn: Asymptotics of the chromatic index for multigraphs,J. Combin. Theory Ser. B,68 (1996), 233–254.

    Google Scholar 

  12. [12]

    J. Kahn: Asymptotics of the list-chromatic index for multigraphs, submitted.

  13. [13]

    J. Kahn: A normal law for matchings, submitted.

  14. [14]

    J. Kahn andP. M. Kayll: Fractional v. integral covers in hypergraphs of bounded edge size,J. Combin. Theory Ser. A,78 (1997), 199–235.

    Google Scholar 

  15. [15]

    J. Kahn andJ.-H. Kim: Random matchings in regular graphs,Combinatorica, to appear.

  16. [16]

    P. M. Kayll:Asymptotically Good Covers in Hypergraphs, Dissertation, Rutgers University, New Brunswick, NJ, 1994.

    Google Scholar 

  17. [17]

    P. M. Kayll: Asymptotically good covers in hypergraphs,Diss. Summ. Math.,1 (1996), 9–16.

    Google Scholar 

  18. [18]

    P. M. Kayll: “Normal” distributions on matchings in a multigraph: overview with applications,Congr. Numer.,107 (1995), 179–191.

    Google Scholar 

  19. [19]

    H. Kunz: Location of the zeros of the partition function for some classical lattice systems,Phys. Lett.,32A (1970), 311–312.

    Google Scholar 

  20. [20]

    C. W. Lee: Some recent results on convex polytopes,Contemporary Mathematics,114 (1990), 3–19.

    Google Scholar 

  21. [21]

    C. W. Lee: Convex polytopes, the moment map, and canonical convex combinations, manuscript, 1994.

  22. [22]

    G. M. Louth:Stochastic Networks: Complexity, Dependence and Routing, Dissertation, Cambridge University, 1990.

  23. [23]

    L. Lovász andM. D. Plummer:Matching Theory, North-Holland, New York, 1986.

    Google Scholar 

  24. [24]

    J. R. Munkres:Elements of Algebraic Topology, Benjamin/Cummings, Menlo Park, 1984.

    Google Scholar 

  25. [25]

    Y. Rabinovich, A. Sinclair andA. Wigderson: Quadratic Dynamical Systems (Preliminary Version), in:Proc. 33rd IEEE Symposium on Foundations of Computer Science, pp. 304–313, 1992.

  26. [26]

    A. Schrijver:Theory of Linear and Integer Programming, Wiley, New York, 1986.

    Google Scholar 

  27. [27]

    J. Spencer:Ten Lectures on the Probabilistic Method, SIAM, Philadelphia, 1993.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

Supported in part by NSF

This work forms part of the author's doctoral dissertation [16]; see also [17]. The author gratefully acknowledges NSERC for partial support in the form of a 1967 Science and Engineering Scholarship.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kahn, J., Kayll, P.M. On the stochastic independence properties of hard-core distributions. Combinatorica 17, 369–391 (1997). https://doi.org/10.1007/BF01215919

Download citation

Mathematics Subject Classification (1991)

  • 05C70
  • 05C65
  • 60C05
  • 52B12
  • 82B20