  We study the asymptotic expansion of the determinant of the graph Laplacian
associated to discretizations of a half-translation 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 sum of
cycle-rooted spanning forests weighted by the monodromy of the connection of
the unitary vector bundle, to the corresponding zeta-regularized determinants.
  As one application, by combining our result with a recent work of
Kassel-Kenyon, modulo some universal topological constants, we give an explicit
formula for the limit of the probability that a cycle-rooted spanning forest
with non-contractible loops, sampled uniformly on discretizations approaching a
given surface, induces the given lamination by its cycles. We also calculate an
explicit value for the limit of certain topological observables on the
associated loop measures.
