I am surprised no one mentioned the work of Alan Sokal and coworkers on precisely this issue of weighted enumeration of spanning forests which is related to the $q\rightarrow 0$ limit of the Potts model as well as the multivariate Tutte polynomial.
A determinant expression corresponds to a Fermionic (Grassmann/Berezin) integral with a quadratic `action' in the exponential. There is an analogue of the matrix-tree theorem for spanning forests with a Fermionic integral with quartic action see: Fermionic field theory for trees and forests which appeared in PRL. Follow ups such as Grassmann Integral Representation for Spanning Hyperforests can be found by looking at Sokal's papers on arxiv.
Note that there was a whole semester at the Isaac Newton Institute revolving around this topic: Combinatorial identities and their applications in statistical mechanics. One can even watch the videos of the talks. The one perhaps most relevant to this question is the talk by Andrea Sportiello in the fourth workshop.
