  This paper completes the comprehensive study of the dimer model on infinite
minimal graphs with Fock's weights [arXiv:1503.00289] initiated in
[arXiv:2007.14699]: the latter article dealt with the elliptic case, i.e.,
models whose associated spectral curve is of genus one, while the present work
applies to models of arbitrary genus. This provides a far-reaching extension of
the genus zero results of [arXiv:math-ph/0202018, arXiv:math/0311062], from
isoradial graphs with critical weights to minimal graphs with weights defining
an arbitrary spectral data. For any minimal graph with Fock's weights, we give
an explicit local expression for a two-parameter family of inverses of the
associated Kasteleyn operator. In the periodic case, this allows us to prove
local formulas for all ergodic Gibbs measures, thus providing an alternative
description of the measures constructed in [arXiv:math-ph/0311005]. We also
compute the corresponding slopes, exhibit an explicit parametrization of the
spectral curve, identify the divisor of a vertex, and build on
[arXiv:math/0311062, arXiv:1107.5588] to establish a correspondence between
Fock's models on periodic minimal graphs and Harnack curves endowed with a
standard divisor.
