Corpus ID: 235458602

Independent sets of a given size and structure in the hypercube

  title={Independent sets of a given size and structure in the hypercube},
  author={Matthew Jenssen and Will Perkins and Aditya Potukuchi},
Abstract. We determine the asymptotics of the number of independent sets of size ⌊β2⌋ in the discrete hypercube Qd = {0, 1} for any fixed β ∈ [0, 1] as d → ∞, extending a result of Galvin for β ∈ [1−1/ √ 2, 1]. Moreover, we prove a multivariate local central limit theorem for structural features of independent sets in Qd drawn according to the hard core model at any fixed fugacity λ > 0. In proving these results we develop several general tools for performing combinatorial enumeration using… Expand
3 Citations
Approximate counting and sampling via local central limit theorems
A new local central limit theorem for the hard-core model that applies to all fugacities below λc(∆), the uniqueness threshold on the infinite ∆-regular tree is proved. Expand
Enumerating independent sets in Abelian Cayley graphs
We show that any connected Cayley graph Γ on an Abelian group of order 2n and degree Ω̃(log n) has at most 2(1+ o(1)) independent sets. This bound is tight up to to the o(1) term when Γ is bipartite.Expand
Approximately counting independent sets in bipartite graphs via graph containers
By implementing algorithmic versions of Sapozhenko’s graph container methods, we give new algorithms for approximating the number of independent sets in bipartite graphs. Our first algorithm appliesExpand


Independent sets in the hypercube revisited
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 combineExpand
A Threshold Phenomenon for Random Independent Sets in the Discrete Hypercube
  • David Galvin
  • Computer Science, Mathematics
  • Combinatorics, Probability and Computing
  • 2010
The asymptotics of Zλ(Qd), the sum over independent sets in Qd with each set I given weight λ|I| are obtained, and nearly matching upper and lower bounds for $\gl \leq \sqrt{2}-1$, extending work of Korshunov and Sapozhenko. Expand
The number of graphs and a random graph with a given degree sequence
An asymptotic formula is obtained approximating the number of graphs within a relative error which approaches 0 as n grows and it is proved that the structure of a random graph with a given tame degree sequence D is well described by a certain maximum entropy matrix computed from D. Expand
A robust quantitative local central limit theorem with applications to enumerative combinatorics and random combinatorial structures
A useful heuristic in the understanding of large random combinatorial structures is the Arratia-Tavare principle, which describes an approximation to the joint distribution of component-sizes usingExpand
The structure of random partitions of large integers
Random partitions of integers are treated in the case where all partitions of an integer are assumed to have the same probability. The focus is on limit theorems as the number being partitionedExpand
The independent set sequence of regular bipartite graphs
This work considers the independent set sequence of finite regular bipartite graphs, and graphs obtained from these by percolation (independent deletion of edges), and obtains partial unimodality results in these cases. Expand
Complex martingales and asymptotic enumeration
This work considerably strengthens existing results on the relationship between random graphs or bipartite graphs with specified degrees and the so-called $\beta$-model of random graphs with independent edges, which is equivalent to the Rasch model in the bipartites case. Expand
Maximum entropy Gaussian approximations for the number of integer points and volumes of polytopes
A maximum entropy approach for computing volumes and counting integer points in polyhedra by solving a certain entropy maximization problem and obtaining asymptotic formulas for volumes of multi-index transportation polytopes and for the number ofMulti-way contingency tables. Expand
Maximum entropy and integer partitions
We derive asymptotic formulas for the number of integer partitions with given sums of jth powers of the parts for j belonging to a finite, non-empty set J ⊂ N. The method we use is based on theExpand
Independent Process Approximations for Random Combinatorial Structures
Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weightedExpand