Integrable spin, vertex and loop models

Abstract : This thesis deals with several problems in statistical mechanics, related to the study of integrable models. In these models, some particular symmetries, like those expressed by the Yang-Baxter equations or "star-triangle'' transformations, lead to the existence of exact formulas for observables of interest. In a first part, we study the star-triangle transformation of the Ising model, recast into a singe polynomial evolution equation by Kashaev. We show that this evolution creates combinatorial objects: $C^{(1)}_2$ loop models. We show some limit shapes results and compute the free energy of these loop models. In a second part, we develop the study of the "eight-vertex'' model, that generalises ice models. In the free-fermion regime, we translate these models into dimers on a bipartite graph, and use the strong integrability structures of these. We deduce the construction of Gibbs measures and correlations in infinite volume, in particular for $Z$-invariant regimes on isoradial graphs. Finally, we suggest interpretations of the Yang-Baxter equations in discrete geometry, via particular embeddings of graphs.
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Paul Melotti. Integrable spin, vertex and loop models. Probability [math.PR]. Sorbonne université, 2019. English. ⟨tel-02277485⟩

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