Lower bounds for multivariate independence polynomials and their generalisations

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 $Z_G(\lambda_1,\dots,\lambda_n) := \sum_{I\in\mathcal{I}(G)} \prod_{v\in I}\lambda_v$, 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 $\lambda_1,\dots,\lambda_n\geq 0$, $$Z_G(\lambda_1,\dots,\lambda_n) \geq \prod_{i=1}^n (1+(d_i+1)\lambda_i)^{1/(d_i+1)},$$ where $d_1,\dots,d_n$ is the degree sequence of $G$. This generalises a result of Sah, Sawhney, Stoner, and Zhao, who proved the univariate case $\lambda_1=\dots=\lambda_n=\lambda$. We further conjecture that our inequality should generalise to other antiferromagnetic models and give some evidence in support of it. In particular, for $\lambda_i,\mu_i\geq 0$, $1\leq i\leq n$, we obtain a stronger inequality $$\sum_{\substack{I,J\in \mathcal{I}(G) \\ I\cap J=\emptyset}} \prod_{v\in I}\lambda_v\prod_{u\in J}\mu_u \geq \prod_{i=1}^n \left(1+(d_i+1)(\lambda_i+\mu_i)+d_i(d_i+1)\lambda_i\mu_i\right)^{1/(d_i+1)},$$ which proves our conjecture for a multiaffine generalisation of the semiproper colouring partition function with two proper colours. Our key technical steps for both theorems are obtained by using a custom mathematical research agent built on top of Gemini Deep Think, which can be seen as a benchmark demonstrating that the current state-of-the-art language models can, in part, assist with mathematical research.