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arXiv · 2610.32527

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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Weiyuan Zhang, Kexiang Xu. 2026-10-23. The upper bounds on occupancy fraction and variance fraction for linear regular uniform hypergraphs. https://arxiv.org/abs/2610.32527

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