  The dimer model is an exactly solvable model of planar statistical mechanics.
In its critical phase, various aspects of its scaling limit are known to be
described by the Gaussian free field. For periodic graphs, criticality is an
algebraic condition on the spectral curve of the model, determined by the edge
weights; isoradial graphs provide another class of critical dimer models, in
which the edge weights are determined by the local geometry. In the present
article, we consider another class of graphs: general Temperleyan graphs, i.e.
graphs arising in the (generalized) Temperley bijection between spanning trees
and dimer models. Building in particular on Forman's formula and
representations of Laplacian determinants in terms of Poisson operators, and
under a minimal assumption - viz. that the underlying random walk converges to
Brownian motion - we show that the natural topological observable on
macroscopic tori converges in law to its universal limit, i.e. the law of the
periods of the dimer height function converges to that of the periods of a
compactified free field.
