# 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.
Keywords :
Domain :

Cited literature [209 references]

https://tel.archives-ouvertes.fr/tel-02277485
Contributor : Paul Melotti <>
Submitted on : Tuesday, September 3, 2019 - 4:01:30 PM
Last modification on : Friday, May 29, 2020 - 3:59:55 PM
Document(s) archivé(s) le : Wednesday, February 5, 2020 - 8:00:14 PM

### File

these_melotti_finale.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : tel-02277485, version 1

### Citation

Paul Melotti. Integrable spin, vertex and loop models. Probability [math.PR]. Sorbonne université, 2019. English. ⟨tel-02277485⟩

Record views