A Threshold Phenomenon for Random Independent Sets in the Discrete Hypercube

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 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∩E|, |I∩O|} = 0 almost surely, where E andO are the bipartition classes of Qd, whereas for λ < 1, min{|I ∩ E|, |I ∩ O|} is almost surely exponential in 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 λ > √ 2− 1, and nearly matching upper and lower bounds for λ ≤ √ 2−1, extending Korshunov and Sapozhenko’s asymptotic estimation at λ = 1. We apply our estimates to the problem of counting the number of independent sets in Qd which contain a fixed proportion of the vertices, obtaining asymptotics when that proportion is greater than 1− 1/ √ 2, and near matching upper and lower bounds otherwise. 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 hardcore distribution with parameter λ, conditioned on a particular v ∈ E being in I, then the probability that another vertex w is in I is o(1) for w ∈ O but Ω(1) for w ∈ E . ∗Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame IN 46556; [email protected]. 1