Universality for the random-cluster model on isoradial graphs

Abstract

We show that the canonical random-cluster measure associated to isoradial graphs is critical for all q 1. Additionally, we prove that the phase transition of the model is of the same type on all isoradial graphs: continuous for 1 q 4 and discontinuous for q > 4. For 1 q 4, the arm exponents (assuming their existence) are shown to be the same for all isoradial graphs. In particular, these properties also hold on the triangular and hexagonal lattices. Our results also include the limiting case of quantum random-cluster models in 1 + 1 dimensions.The random-cluster model is a dependent percolation model that generalises Bernoulli percolation. It was introduced by Fortuin and Kasteleyn in [18] to unify percolation theory, electrical network theory and the Potts model. The spin correlations of the Potts model get rephrased as cluster connectivity properties of its random-cluster representation, and can therefore be studied using probabilistic techniques coming from percolation theory.The random-cluster model on the square lattice has been the object of intense study in the past few decades. A duality relation enables to prove that the model undergoes a phase transition at the self-dual value p c = √ q 1+ √ q of the edge-parameter [3] (see also [13,14,15]). It can also be proved that the distribution of the size of finite clusters has exponential tails when the model is non-critical. Also, the critical phase is now fairly well understood: the phase transition of the model is continuous if the cluster-weight belongs to [1,4] [16] and discontinuous if it is greater than 4 [12]. When the cluster-weight is equal to 2, the random-cluster model is coupled with the Ising model, and is known to be conformally invariant [37,8] (we also refer to [17] for a review).A general challenge in statistical physics consists in understanding universality, i.e., that the behaviour of a certain model is not affected by small modifications of its definition. This is closely related to the so-called conformal invariance of scaling limits: when we scale out the model at criticality, the resulting limit should be preserved under conformal transformations, including translations, rotations and Möbius maps.The goal of this paper is to prove a form of universality for a certain class of random-cluster models. Specifically we aim to transfer results obtained for the square lattice to a larger class of graphs called isoradial graphs, i.e., planar graphs embedded in the plane in such a way that every face is inscribed in a circle of radius one. A specific random-cluster model is associated to each such graph, where the edge-weight of every edge is an explicit function of its length. Moreover, the edge-weight is expected to compensate the inhomogeneity of the embedding and render the model conformally invariant in the limit.Isoradial graphs were introduced by Duffin in [10] in the context of discrete complex analysis, and later appeared in the physics literature in the work of Baxter [1], where they are called Z-invariant graphs. The term isoradial wa...

Dimerization and Néel Order in Different Quantum Spin Chains Through a Shared Loop Representation

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