Cube moves for s-embeddings and α-realizations
Annales de l’Institut Henri Poincaré D, Tome 10 (2023) no. 4, pp. 781-817.
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Chelkak introduced s-embeddings as tilings by tangential quads which provide the right setting to study the Ising model with arbitrary coupling constants on arbitrary planar graphs. We prove the existence and uniqueness of a local transformation for s-embeddings called the cube move, which consists in flipping three quadrilaterals in such a way that the resulting tiling is also in the class of s-embeddings. In passing, we give a new and simpler formula for the change in coupling constants for the Ising star-triangle transformation which is conjugated to the cube move for s-embeddings. We introduce more generally the class of α-embeddings as tilings of a portion of the plane by quadrilaterals such that the side lengths of each quadrilateral ABCD satisfy the relation AB α +CD α =AD α +BC α , providing a common generalization for harmonic embeddings adapted to the study of resistor networks (α=2) and for s-embeddings (α=1). We investigate existence and uniqueness properties of the cube move for these α-embeddings.

Publié le :
DOI : 10.4171/aihpd/163
Classification : 82-XX, 05-XX, 16-XX
Mots-clés : Star-triangle, $Y$-delta, Yang–Baxter equations, cube flips, cube moves, $s$-embedding, Ising model, embedding, harmonic embedding, integrable system 
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     title = {Cube moves for $s$-embeddings and $\alpha$-realizations},
     journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
     pages = {781--817},
     volume = {10},
     number = {4},
     year = {2023},
     doi = {10.4171/aihpd/163},
     mrnumber = {4653797},
     zbl = {1530.82007},
     language = {en},
     url = {http://archive.numdam.org/articles/10.4171/aihpd/163/}
}
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Melotti, Paul; Ramassamy, Sanjay; Thévenin, Paul. Cube moves for $s$-embeddings and $\alpha$-realizations. Annales de l’Institut Henri Poincaré D, Tome 10 (2023) no. 4, pp. 781-817. doi : 10.4171/aihpd/163. http://archive.numdam.org/articles/10.4171/aihpd/163/

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