混合配分函数恰为指数有界边连通秩的图参数
Mixed partition functions are exactly the graph parameters of exponentially bounded edge-connection rank
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中文总结 AI 辅助
该论文证明Regts与Sevenster的猜想，建立指数有界边连通秩的图参数与混合配分函数的等价性，构造对应连通范畴并结合Deligne定理得到超张量网络，附录给出幂等迹步骤的独立证明。
中文摘要 AI 辅助
我们证明了Regts与Sevenster的猜想：满足f(∅)=1的复值图参数f具有指数有界边连通秩当且仅当它是混合配分函数；此外，该模型的偶颜色与奇颜色数量可根据秩界显式确定。我们从f出发构造了一个连通范畴，这是一个刚性对称的ℂ线性幺半范畴，其态射空间以连通秩为维数，且其迹配对是非退化的。秩假设迫使张量适度增长，而Etingof与Penneys的最新定理表明，每个幂等自同态的迹为零；结合迹配对的非退化性，该范畴是半单的，Deligne定理提供了一个忠实对称张量函子到有限维超向量空间。我们随后将所得超张量网络与Regts-Sevenster模型完全对应，即其欧拉子图展开及每个费米子回路的符号为-1。附录给出了幂等迹步骤的独立直接证明，表明在满足End(1)=ℂ的刚性对称ℂ线性范畴中，指数有界自同态增长使每个自同态的迹ζ函数为有理函数，且有显式次数界。
英文摘要
We prove a conjecture of Regts and Sevenster: a complex-valued graph parameter $f$ with $f(\emptyset)=1$ has exponentially bounded edge-connection rank if and only if it is a mixed partition function. The bound is exact: for a real number $R\ge 1$, the connection ranks satisfy $\operatorname{rk} M_{f,t}\le R^t$ for all $t\ge 0$ if and only if $f$ has a model on a super vector space $\mathbb{C}^{k|2\ell}$ with $k+2\ell\le R$. Consequently the base of exponential growth of the connection ranks is the least number of colours of a model, the two dimensions of a minimal model are determined by $f$, and the parameters with a model on a prescribed $\mathbb{C}^{k|2\ell}$ are characterised. The proof organises fragments modulo the connection kernel into a rigid symmetric tensor category whose morphism spaces have the connection ranks as dimensions; the rank hypothesis and an argument of Schrijver make its additive idempotent completion semisimple, Deligne's theorem provides a fibre functor to super vector spaces, and the resulting super tensor network is identified with the Regts-Sevenster model exactly, circuit signs included. An appendix shows that in a rigid symmetric $\mathbb{C}$-linear category with $\mathrm{End}(\mathbf{1})=\mathbb{C}$, exponentially bounded endomorphism growth makes the trace zeta function of every endomorphism rational, with explicit degree bounds.
