AI 中文总结
该研究构建了超图上统计力学模型的严谨数学框架，将Tutte多项式推广至超图，定义了三类超图模型及对应多拟阵，为高阶自旋模型提供了统一理论基础。
AI 中文摘要
我们为超图上的统计力学模型构建了严谨的数学框架，并给出了将经典的Potts模型配分函数与Tutte多项式之间的关联从图推广到超图的条件。我们定义了超图模型及其对应的配分函数，并将其扩展到由相互作用函数族诱导的特殊模型类。对于映射到{0,1}的布尔相互作用函数族，我们证明其配分函数由组合秩函数决定，并建立了超图删除-收缩递推关系的充分条件，以及该秩函数定义多拟阵的充分条件。我们通过将该理论应用于三类超图相互作用函数族来加以说明：奇偶伊辛模型（Parity Ising）、Δ-Potts模型（Delta Potts）和与伊辛模型（And Ising）。这些诱导的超图模型彼此并不同构，但前两类可归约为相同的图伊辛模型。我们为这些模型识别出与超图自然关联的三类多拟阵：分别是在域F₂上的关联矩阵的二元拟阵、超图多拟阵以及布尔多拟阵。对于图而言，前两类可归约为经典图拟阵，其配分函数可还原为多元Tutte多项式。因此，Tutte多项式至少有两种不同的超图推广形式，均满足删除-收缩递推关系：一种是超图关联矩阵的二元拟阵的Tutte多项式，另一种是超图多拟阵的庞加莱多项式的多元版本。与伊辛模型的配分函数同样是多元庞加莱多项式，此处对应布尔多拟阵。这些例子阐明了超图模型的范围要广泛得多，并强调了统一理论的必要性。
英文摘要
We develop a rigorous mathematical framework for statistical mechanics models on hypergraphs, and give conditions for lifting the classical connection between Potts model partition functions and the Tutte polynomial from graphs to hypergraphs. We define hypergraphical models and their associated partition functions, and extend these to special classes of models induced by families of interaction functions. For boolean interaction families, whose interaction functions map to $\{0,1\}$, we show that the partition function is determined by a combinatorial rank function, and we establish sufficient conditions for a hypergraph deletion-contraction recurrence and for when the rank function defines a polymatroid. We illustrate the theory by applying it to three hypergraph interaction families: Parity Ising, Delta Potts, and And Ising. The induced hypergraphical models are not isomorphic to each other, but the first two reduce to the same graphical Ising models. We identify three polymatroids naturally associated with hypergraphs for these models: respectively, the binary matroid of the incidence matrix over $\mathbb{F}_2$, the hypergraphical polymatroid, and the boolean polymatroid. For graphs, the first two reduce to the classical graphical matroid, and their partition functions recover the multivariate Tutte polynomial. The Tutte polynomial thus admits at least two distinct generalizations for hypergraphs, both satisfying a deletion-contraction recurrence: the Tutte polynomial of the binary matroid of the hypergraph incidence matrix, and a multivariate version of the Poincaré polynomial of the hypergraphical polymatroid. The partition functions of And Ising models are likewise multivariate versions of the Poincaré polynomial, here of the boolean polymatroid. These examples illustrate the much greater range of hypergraphical models and underscore the need for the unifying theory.
