  In the dimer model, a configuration consists of a perfect matching of a fixed
graph. If the underlying graph is planar and bipartite, such a configuration is
associated to a height function. For appropriate "critical" (weighted) graphs,
this height function is known to converge in the fine mesh limit to a Gaussian
free field, following in particular Kenyon's work.
  In the present article, we study the asymptotics of smoothed and local field
observables from the point of view of families of Cauchy-Riemann operators and
their determinants. This allows in particular to obtain a functional invariance
principle for the field; characterise completely the limiting field on toroidal
graphs as a compactified free field; analyse electric correlators; and settle
the Fisher-Stephenson conjecture on monomer correlators.
  The analysis is based on comparing the variation of determinants of families
of (continuous) CR operators with that of their discrete (finite dimensional)
approximations. This relies in turn on estimating precisely inverting kernels,
in particular near singularities. In order to treat correlators of "singular"
local operators, elements of (multiplicatively) multi-valued discrete
holomorphic functions are discussed.
