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Independent sets of a given size and structure in the hypercube

Part of: Limit theorems Graph theory Equilibrium statistical mechanics

Published online by Cambridge University Press:  06 January 2022

Matthew Jenssen ,
Will Perkins Open the ORCID record for Will Perkins [Opens in a new window]  and
Aditya Potukuchi
Matthew Jenssen
Affiliation:
School of Mathematics, University of Birmingham, Birmingham, UK
Will Perkins*
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA
Aditya Potukuchi
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA
*
*Corresponding author. E-mail: william.perkins@gmail.com
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Abstract

We determine the asymptotics of the number of independent sets of size $\lfloor \beta 2^{d-1} \rfloor$ in the discrete hypercube $Q_d = \{0,1\}^d$ for any fixed $\beta \in (0,1)$ as $d \to \infty$ , extending a result of Galvin for $\beta \in (1-1/\sqrt{2},1)$ . Moreover, we prove a multivariate local central limit theorem for structural features of independent sets in $Q_d$ drawn according to the hard-core model at any fixed fugacity $\lambda>0$ . In proving these results we develop several general tools for performing combinatorial enumeration using polymer models and the cluster expansion from statistical physics along with local central limit theorems.

Keywords

independent setshypercubecluster expansionlocal central limit theoremasymptotic enumerationpolymer models

MSC classification

Primary: 05C30: Enumeration in graph theory 05C69: Dominating sets, independent sets, cliques
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 60F05: Central limit and other weak theorems
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 31 , Issue 4 , July 2022 , pp. 702 - 720
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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