Spanning trees, cycle-rooted spanning forests on discretizations of flat surfaces and analytic torsion

We study the asymptotic expansion of the determinant of the graph Laplacian associated to discretizations of a tileable surface endowed with a flat unitary vector bundle. By doing so, over the discretizations, we relate the asymptotic expansion of the number of spanning trees and the partition function of cycle-rooted spanning forests weighted by the monodromy of the unitary connection on the vector bundle, to the corresponding zeta-regularized determinants. As a consequence, we establish open problems 2 and 4, formulated by Kenyon in 2000. The spectral theory on discretizations of flat surfaces, Fourier analysis on discrete square and the analytic methods used in the proof of Ray–Singer conjecture lie in the core of our approach.