Mixed partition functions are exactly the graph parameters of exponentially bounded edge-connection rank

Abstract

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.