Semiclassical limits of quantum partition functions on infinite graphs

We prove that if $H$ denotes the operator corresponding to the canonical Dirichlet form on a possibly locally infinite weighted graph $(X,b,m)$, and if $v:X\to \mathbb{R}$ is such that $H+v/\hbar$ is well-defined as a form sum for all $\hbar >0$, then the quantum partition function $\mathrm{tr}(\mathrm{e}^{-\beta \hbar ( H + v/\hbar)})$ satisfies $$ \mathrm{tr}(\mathrm{e}^{-\beta \hbar ( H + v/\hbar)})\xrightarrow[]{\hbar\to 0+}\sum_{x\in X} \mathrm{e}^{-\beta v(x)} \text{ for all $\beta>0$}, $$ regardless of the fact whether $\mathrm{e}^{-\beta v}$ is apriori summable or not. We also prove natural generalizations of this semiclassical limit to a large class of covariant Schr\"odinger operators that act on sections in Hermitian vector bundle over $(X,m,b)$, a result that particularly applies to magnetic Schr\"odinger operators that are defined on $(X,m,b)$.