Dimers for degenerating families of M-curves

We study dimer models on infinite minimal graphs with Fock's weights for degenerating families of M-curves of any genus based on works of Boutillier-Cimasoni-de Tilière and Bobenko A.I.- Bobenko N.-Suris for a fixed M-curve. We show that these dimer models for families of M-curves behave consistently under their degeneration shrinking real circles to points, and that they can be calculated as power series in the associated deformation parameters which are regarded as the perturbation of Kenyon's critical dimer models.