On expectations and variances in the hard-core model

The hard-core model can be used to understand the numbers of independent sets in graphs in extremal graph theory. The occupancy fraction, defined as the logarithmic derivative of the independence polynomial of a graph, is a key quantity in hard-core model. Davies \textit{et al.} (2017) established an upper bound on the occupancy fraction for $d$-regular graphs, and Perarnau and Perkins (2018) derived a corresponding bound on it for graphs with given girth. Inspired by their work, we provide the tight upper and lower bounds on occupancy fraction in $n$-vertex graphs with independence number $\alpha$, extending the classical results on bounds for independence polynomials. We also prove a relevant conjecture posed by Davies \textit{et al.} (2025) to this topic.