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PreprintarXiv licencearXiv

The upper bounds on occupancy fraction and variance fraction for linear regular uniform hypergraphs

  • Weiyuan Zhang
  • Kexiang Xu
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Abstract

The hard-core model can be used to understand the number of independent sets in graphs in extremal graph theory. The occupancy and variance fractions are the expectation and variance, respectively, of the independent-set size, normalized by the number of vertices. We provide upper bounds on the occupancy and variance fractions of linear d-regular r-uniform hypergraphs for small positive fugacities. In proving these results, we use the cluster expansion and the abstract polymer models, two tools from statistical physics, to analyse auxiliary functions associated with independence polynomials, comparing their low-order coefficients and controlling the remainders uniformly in the number of vertices.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026upper,
  title = {{The upper bounds on occupancy fraction and variance fraction for linear regular uniform hypergraphs}},
  author = {Weiyuan Zhang and Kexiang Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32527},
  eprint = {2610.32527},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32527},
}
RIS
TY  - UNPB
TI  - The upper bounds on occupancy fraction and variance fraction for linear regular uniform hypergraphs
AU  - Weiyuan Zhang
AU  - Kexiang Xu
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32527
UR  - https://arxiv.org/abs/2610.32527
AB  - The hard-core model can be used to understand the number of independent sets in graphs in extremal graph theory. The occupancy and variance fractions are the expectation and variance, respectively, of the independent-set size, normalized by the number of vertices. We provide upper bounds on the occupancy and variance fractions of linear d-regular r-uniform hypergraphs for small positive fugacities. In proving these results, we use the cluster expansion and the abstract polymer models, two tools from statistical physics, to analyse auxiliary functions associated with independence polynomials, comparing their low-order coefficients and controlling the remainders uniformly in the number of vertices.
ER  -