Independent sets in the hypercube revisited

Abstract

We revisit Sapozhenko's classic proof on the asymptotics of the number of independent sets in the discrete hypercube {0, 1} d and Galvin's follow-up work on weighted independent sets. We combine Sapozhenko's graph container methods with the cluster expansion and abstract polymer models, two tools from statistical physics, to obtain considerably sharper asymptotics and detailed probabilistic information about the typical structure of (weighted) independent sets in the hypercube. These results refine those of Korshunov and Sapozhenko and Galvin, and answer several questions of Galvin. * University of Oxford, jenssen@maths.ox.ac.uk. † University of Illinois at Chicago, math@willperkins.org.

Independent sets of a given size and structure in the hypercube

“…Here we show that local central limit theorems work very well in combination with two tools from statistical physics, polymer models and the cluster expansion, which have been used recently in combinatorial enumeration [15,2,13].…”

“…and obtain explicit asymptotics for any fixed β. More generally, the results of [15] and of this paper hold for much smaller λ and β, tending to 0 as d → ∞, as long as λ ≥ C log d/d 1/3 and β > C log d/d 1/3 for an absolute constant C. In this case, however, the asymptotic formulas in Theorem 3 and Theorem 5 become series with a number of terms that grows with d. These series can be used to give an algorithm to approximate Z(λ) and i ⌊βN ⌋ (Q d ) up to a (1 + ε) multiplicative factor in time polynomial in 1/ε and N (an FPTAS in the language of approximate counting; see e.g. [14] for such an algorithm for independent sets in expander graphs).…”

“…In practice, we do not work with the hard-core model directly, but with an approximating measure derived from a polymer model which describes the distribution on defects in an independent set from the hard-core model (see Section 2). This polymer model was introduced in [15].…”

“…In [15] the authors prove a multivariate central limit theorem for the number of defects of different types in the polymer model. In Section 4, we establish a multivariate local central limit theorem for this polymer model which allows us to refine Theorems 5 and 6 further still.…”

“…Recently, the first two authors found the asymptotics of Z(λ) for all fixed λ > 0 [15]. The asymptotic formula takes into account defects of arbitrary, but constant size.…”