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Journal Articles Probability and Mathematical Physics Year : 2023

Minimal bipartite dimers and higher genus Harnack curves

Abstract

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 give local formulas for all ergodic Gibbs measures, thus providing an alternative description of the measures constructed in [arXiv:math-ph/0311005]. We also give formulas for the corresponding slopes, an explicit parametrisation of the spectral curve, and build on [arXiv:math/0311062, arXiv:1107.5588] to establish a correspondence between Fock's models on periodic minimal graphs and Harnack curves.

Dates and versions

hal-03502141 , version 1 (24-12-2021)

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Cite

Cédric Boutillier, David Cimasoni, Béatrice de Tilière. Minimal bipartite dimers and higher genus Harnack curves. Probability and Mathematical Physics, 2023, 4 (1), pp.151-208. ⟨10.2140/pmp.2023.4.151⟩. ⟨hal-03502141⟩
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