This paper completes the comprehensive study of the dimer model on infinite
minimal graphs with Fock’s weights (arXiv:1503.00289 (2015)) initiated
in Comm. Math. Phys. (2023): the latter article dealt with the elliptic case, i.e.,
models whose associated spectral curve is of genus 1, while the present work
applies to models of arbitrary genus. This provides a far-reaching extension of
the genus 0 results of Kenyon (Invent. Math.150:2 (2002), 409–439) and
Kenyon and Okounkov (Duke Math. J.131:3 (2006), 499–524), 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
Ann. of Math.163:3 (2006),
1019–1056. We also compute the corresponding slopes, exhibit an explicit
parametrization of the spectral curve, identify the divisor of a vertex, and
build on
Ann. Sci.Éc. Norm. Supér.46:5 (2013), 747–813 and
DukeMath. J.131:3 (2006), 499–524 to establish a correspondence between Fock’s models
on periodic minimal graphs and Harnack curves endowed with a standard
divisor.
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