The renormalisation group flow predicts that scaling limits of critical statistical mechanics models are described by conformal field theories. In particular, they should have conformal symmetries and not depend on the exact microscopic details of the model, but rather only depend on a so-called universality class. In models with suitable integrability, this conformal covariance has even been proven in a large amount of cases, e.g. the uniform spanning tree, dimers and the Ising model. In this talk, we focus on the general critical random-cluster model, for which less integrability is available. As such, proving conformal invariance from the available parafermionic observables is difficult, so we follow the other route of proving that microscopic details do not matter: Using the Yang-Baxter Equation for the random-cluster model with isoradial weights due to Kenyon and the techniqes from the proof of rotational invariance on the square lattice due to Duminil-Copin, Kozlowski, Krachun, Manolescu and Oulamara, we prove a universality result across the class of doubly-periodic isoradial graphs.
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