On the Variance Fraction of the Hard-Core Model on Graphs with Bounded Maximum Degree

Abstract

The hard-core model can be used to understand the number of independent sets in graphs in extremal graph theory. The occupancy fraction, defined by Davies \textit{et al.} in 2017 as the logarithmic derivative of the independence polynomial of a graph, is a key quantity in the hard-core model. The variance fraction, introduced by Davies \textit{et al.} in 2025, is defined as the derivative of the occupancy fraction with respect to the logarithm of the fugacity. Since the occupancy fraction can be obtained by integrating the variance fraction with respect to the logarithm of the fugacity, bounding the variance fraction yields the corresponding bounds on the occupancy fraction. Moreover, the occupancy fraction correlates, in quantity, to the independence polynomial. In this note we provide two lower bounds on the variance fraction, proving the conjecture by Davies \textit{et al.} in 2025, for graphs with bounded maximum degree and for graphs with $n$ vertices, respectively. We also derive lower bounds for other graph classes, including graphs with a given edge chromatic number, $d$-regular graphs, and triangle-free graphs with bounded maximum degree.