Hermite矩阵上配分函数的复杂性
The Complexity of Partition Functions on Hermitian Matrices
摘要：
统计物理学中某些类别的“自旋玻璃”模型的配分函数与组合图不变量有很强的联系。也被称为同态函数，它们允许表示许多这样的不变量，例如，一个图的独立集的数量或无处为零的k-流的数量。贡献的配分函数的复杂性的最新发展，我们研究的复杂性配分函数与复杂的价值观。这些函数通常由方阵A确定，Goldberg、Grohe、Jerrum和Thurley证明，对于每个实值对称矩阵，相应的配分函数要么是多项式时间可计算的，要么是#P-hard的。推广这一结果，我们给出了一个完整的描述的复杂性的划分功能定义的厄米特矩阵。这些也可以分为多项式时间可计算的和#P-hard的。虽然多项式时间可计算性的标准是不能描述在一个单一的行，我们给它一个明确的帐户与阿贝尔群的结构。
英文摘要：
Partition functions of certain classes of "spin glass" models in statistical physics show strong connections to combinatorial graph invariants. Also known as homomorphism functions they allow for the representation of many such invariants, for example, the number of independent sets of a graph or the number nowhere zero k-flows. Contributing to recent developments on the complexity of partition functions we study the complexity of partition functions with complex values. These functions are usually determined by a square matrix A and it was shown by Goldberg, Grohe, Jerrum, and Thurley that for each real-valued symmetric matrix, the corresponding partition function is either polynomial time computable or #P-hard. Extending this result, we give a complete description of the complexity of partition functions definable by Hermitian matrices. These can also be classified into polynomial time computable and #P-hard ones. Although the criterion for polynomial time computability is not describable in a single line, we give a clear account of it in terms of structures associated with Abelian groups.
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