发表机构
Universiteit van Amsterdam（阿姆斯特丹大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究递归图序列上硬核模型的独立多项式零点，证明在度有界且标记顶点距离发散时零点避开非负实轴，从而由李-杨理论得出无相变，方法基于有理映射的动力学分析。
AI 中文摘要
我们研究了递归定义的图序列 $(G_n)_{n\geq0}$ 上的硬核模型，其中每个图中的标记顶点数固定为 $k\geq 1$。序列中的下一个图通过取前一个图的固定数量 $m\geq 2$ 个副本，根据固定规则识别某些标记顶点来连接这些副本，然后根据固定规则在所得图中选择 $k$ 个标记顶点来构造。此类序列的例子包括谢尔宾斯基垫片图、层级晶格等。我们证明了，当图 $G_n$ 的顶点度一致有界且 $G_n$ 中标记顶点之间的距离发散时，单变量独立多项式 $Z_{G_n}(\lambda)$ 的复零点避开非负实轴的一个邻域。根据李-杨理论，这意味着在这些递归图序列上的硬核模型不会发生相变，与起始图 $G_0$ 无关。证明依赖于对由图像递归算子诱导的 $(2^k-1)$ 维复射影空间上的单参数有理映射族 $F_\lambda$ 的动力学性质的研究。本文发展的动力学框架可以自然地扩展到统计力学中的其他经典模型（如伊辛或波茨模型）以及更一般的图递归概念。
英文摘要
We study the hard-core model on recursively defined sequences $(G_n)_{n\geq0}$ of graphs with a fixed number $k\geq 1$ of labeled vertices in each graph. The next graph in the sequence is constructed by taking a fixed number $m\geq 2$ of copies of the previous graph, connecting these copies by identifying some labeled vertices according to a fixed rule, and afterward choosing $k$ labeled vertices in the resulting graph, again in accordance with a fixed rule. Examples of such sequences include the Sierpiński gasket graphs, hierarchical lattices, and many more. We prove that, when the vertex degrees of the graphs $G_n$ are uniformly bounded and the distances between the labeled vertices in $G_n$ diverge, the complex zeros of the univariate independence polynomials $Z_{G_n}(λ)$ avoid a neighborhood of the non-negative real axis. By the Lee--Yang theory this implies that no phase transitions occur for the hard-core model on these recursive sequences of graphs, independently of the starting graph $G_0$. The proof relies on the study of the dynamical properties of a one-parameter family of rational maps $F_λ$ on the $(2^k-1)$-dimensional complex projective space induced by the graph recursion operator. The dynamical framework developed in this paper can be naturally extended to other classical models in statistical mechanics (such as the Ising or Potts models) and to more general notions of graph recursions.
Comments58 pages, 4 figures
