Elliptic dimers on minimal graphs and genus 1 Harnack curves
Boutillier, Cédric; Cimasoni, David; de Tilière, Béatrice (2022), Elliptic dimers on minimal graphs and genus 1 Harnack curves. https://basepub.dauphine.psl.eu/handle/123456789/23977
TypeDocument de travail / Working paper
External document linkhttps://hal.science/hal-02908609
Series titleCahier de recherche CEREMADE, Université Paris Dauphine-PSL
MetadataShow full item record
Laboratoire de Probabilités, Statistique et Modélisation [LPSM (UMR_8001)]
Section de mathématiques [Genève]
de Tilière, Béatrice
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)This paper provides a comprehensive study of the dimer model on infinite minimal graphs with Fock's elliptic weights [Foc15]. Specific instances of such models were studied in [BdTR17, BdTR18, dT17]; we now handle the general genus 1 case, thus proving a non-trivial extension of the genus 0 results of [Ken02, KO06] on isora-dial critical models. We give an explicit local expression for a two-parameter family of inverses of the Kasteleyn operator with no periodicity assumption on the underlying graph. When the minimal graph satisfies a natural condition, we construct a family of dimer Gibbs measures from these inverses, and describe the phase diagram of the model by deriving asymptotics of correlations in each phase. In the Z 2-periodic case, this gives an alternative description of the full set of ergodic Gibbs measures constructed in [KOS06]. We also establish a correspondence between elliptic dimer models on periodic minimal graphs and Harnack curves of genus 1. Finally, we show that a bipartite dimer model is invariant under the shrinking/expanding of 2-valent vertices and spider moves if and only if the associated Kasteleyn coefficients are antisymmetric and satisfy Fay's trisecant identity.
Subjects / Keywordsdimers; minimal graphs; elliptic curves; theta functions; determinantal processes; Gibbs measures
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