Loop-Cluster Coupling and Algorithm for Classical Statistical Models

The q-state Potts model plays a major role in the study of critical phenomena, mostly thanks to the different representations one can use for its description, the spin, Fortuin-Kasteleyn (FK) and loop representations. Historically, a coupling between the FK and spin representations was derived and was successfully and extensively used for theoretical and numerical purposes for forty years. In the absence of any coupling with the other representations, the loop one has still to be considered independently, whereas it allows direct access to the magnetic susceptibility or correlation quantities. Following an approach based on the decomposition of random clusters into loop ones, our work now completes the construction of this missing theoretical and numerical link and ties together the three representations by the introduction of a joint-model between the FK and loop ones.