TLDR
在统计物理中，多元硬核模型描述一个粒子系统，每个粒子拥有各自的逸度。用图论语言表述，该模型的配分函数对应多元独立多项式，即独立多项式的多重仿射推广，定义为 $Z_G(λ_1,\dots,λ_n) := \sum_{I\in\mathcal{I}(G)} \prod_{v\in I}λ_v$，其中 $\mathcal{I}(G)$ 表示 $[n]:=\{1,2,\dots,n\}$ 上图 $G$ 的所有独立集。我们证明对于 $[n]$ 上的每个简单图 $G$ 以及 $λ_1,\dots,λ_n\geq 0$，\[ Z_G(λ_1,\dots,In statistical physics, the multivariate hard-core model describes a system of particles, each of which receives its own fugacity. In graph-theoretic language, the partition function of the model translates to the multivariate independence polynomial, i.e., the multiaffine generalisation of the independence polynomial, defined by $Z_G(λ_1,\dots,λ_n) := \sum_{I\in\mathcal{I}(G)} \prod_{v\in I}λ_v$, where $\mathcal{I}(G)$ denotes the set of all independent sets in a graph $G$ on $[n]:=\{1,2,\dots,n\}$. We prove that for every simple graph $G$ on $[n]$ and $λ_1,\dots,λ_n\geq 0$, \[ Z_G(λ_1,\dots,