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A Threshold Phenomenon for Random Independent Sets in the Discrete Hypercube

Published online by Cambridge University Press:  02 July 2010

DAVID GALVIN
DAVID GALVIN*
Affiliation:
Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, IN 46556, USA (e-mail: dgalvin1@nd.edu)
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Abstract

Let I be an independent set drawn from the discrete d-dimensional hypercube Qd = {0, 1}d according to the hard-core distribution with parameter λ > 0 (that is, the distribution in which each independent set I is chosen with probability proportional to λ|I|). We show a sharp transition around λ = 1 in the appearance of I: for λ > 1, min{|I ∩ Ɛ|, |I ∩ |} = 0 asymptotically almost surely, where Ɛ and are the bipartition classes of Qd, whereas for λ < 1, min{|I ∩ Ɛ|, |I ∩ |} is asymptotically almost surely exponential in d. The transition occurs in an interval whose length is of order 1/d.

A key step in the proof is an estimation of Zλ(Qd), the sum over independent sets in Qd with each set I given weight λ|I| (a.k.a. the hard-core partition function). We obtain the asymptotics of Zλ(Qd) for , and nearly matching upper and lower bounds for , extending work of Korshunov and Sapozhenko. These bounds allow us to read off some very specific information about the structure of an independent set drawn according to the hard-core distribution.

We also derive a long-range influence result. For all fixed λ > 0, if I is chosen from the independent sets of Qd according to the hard-core distribution with parameter λ, conditioned on a particular v ∈ Ɛ being in I, then the probability that another vertex w is in I is o(1) for w ∈ but Ω(1) for w ∈ Ɛ.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 1 , January 2011 , pp. 27 - 51
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Bollobás, B. (1998) Modern Graph Theory, Springer, New York.CrossRefGoogle Scholar
[2]Brightwell, G. and Winkler, P. (2002) Hard constraints and the Bethe lattice: Adventures at the interface of combinatorics and statistical physics. In Proc. Int. Congress of Mathematicians Vol. III (Tatsien, Li, ed.), Higher Education Press, Beijing, pp. 605–624.Google Scholar
[3]Galvin, D. (2003) On homomorphisms from the Hamming cube to ℤ. Israel J. Math. 138 189–213.CrossRefGoogle Scholar
[4]Galvin, D. Independent sets of a fixed size in the discrete hypercube. In preparation.Google Scholar
[5]Galvin, D. and Kahn, J. (2004) On phase transition in the hard-core model on ℤd. Combin. Probab. Comput. 13 137–164.Google Scholar
[6]Galvin, D. and Tetali, P. (2006) Slow mixing of the Glauber dynamics for the hard-core model on the Hamming cube. Random Struct. Alg. 28 427–443.CrossRefGoogle Scholar
[7]Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.CrossRefGoogle Scholar
[8]Kahn, J. (2001) An entropy approach to the hard-core model on bipartite graphs. Combin. Probab. Comput. 10 219–237.Google Scholar
[9]Korshunov, A. and Sapozhenko, A. (1983) The number of binary codes with distance 2. Problemy Kibernet. 40 111–130. (In Russian.)Google Scholar
[10]Lovász, L. (1975) On the ratio of optimal integral and fractional covers. Discrete Math. 13 383–390.CrossRefGoogle Scholar
[11]Sapozhenko, A. (1987) On the number of connected subsets with given cardinality of the boundary in bipartite graphs. Metody Diskret. Analiz. 45 42–70. (In Russian.)Google Scholar
[12]Sapozhenko, A. (1989) The number of antichains in ranked partially ordered sets. Diskret. Mat. 1 74–93. (In Russian; translation in Discrete Math. Appl. 1 (1991) 35–58.)Google Scholar
[13]Stein, S. K. (1974) Two combinatorial covering theorems. J. Combin. Theory Ser. A 16 391–397.CrossRefGoogle Scholar