The star-triangle transformation is used to obtain an equivalence extending over the set of all (in)homogeneous bond percolation models on the square, triangular and hexagonal lattices. Among the consequences are box-crossing (RSW) inequalities for such models with parameter-values at which the transformation is valid. This is a step toward proving the universality and conformality of these processes. It implies criticality of such values, thereby providing a new proof of the critical point of inhomogeneous systems. The proofs extend to certain isoradial models to which previous methods do not apply. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2013, Vol. 41, No. 4, 2990. This reprint differs from the original in pagination and typographic detail. 1 to exist. Box-crossing inequalities are useful also in proving scale-invariance for critical percolation in two dimensions (see [8,27]). Consider, for example, a domain D in the plane (with a superimposed lattice L with mesh-size δ) and four points, A, B, C and D distributed anti-clockwise along its boundary. Consider the limit (as δ → 0) of the probability that there exists an open path in D joining the boundary arcs AB and CD. Cardy [9] presented a formula for this limit, and this was proved by Smirnov [27] for the special case of critical site percolation on the triangular lattice. Corresponding statements are expected to hold for other lattices but no proofs are yet known. One may show that, if the underlying measure has the box-crossing property (see Definition 1.2), then such probabilities are bounded uniformly away from 0 and 1 as δ → 0. The box-crossing property plays a significant role in the proof of Cardy's formula, in which one shows the uniform convergence of a certain triplet of discretely harmonic functions to a limiting triplet of harmonic functions. This is obtained in two steps: first, one proves tightness for the family of functions, then one identifies its subsequential limits. Tightness follows by an application of the Arzelà-Ascoli theorem, whose pre-compactness hypothesis is met by the fact that the discretely harmonic functions are uniformly Hölder. The proof of this last fact is via the box-crossing property. A full proof of Cardy's formula may be found in [13], Section 5.7 and [31], Section 2.The structure of the paper is as follows. The necessary notation is introduced in Section 1.2, and our main results stated in Sections 1.3-1.4. The star-triangle transformation is discussed in detail in Section 2, with particular attention to transformations of edge-configurations and open paths. Proofs for inhomogeneous bond percolation on the square, triangular and hexagonal lattices are found in Section 3, and for the highly inhomogeneous models in Section 4. 1.2.Notation. The lattices under study are the square, triangular and hexagonal (or honeycomb) lattices illustrated in Figure 1. The hypotheses and conclusions of this paper may often be expressed in ter...

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In an investigation of percolation on isoradial graphs, we prove the criticality of canonical bond percolation on isoradial embeddings of planar graphs, thus extending celebrated earlier results for homogeneous and inhomogeneous square, triangular, and other lattices. This is achieved via the star-triangle transformation, by transporting the box-crossing property across the family of isoradial graphs. As a consequence, we obtain the universality of these models at the critical point, in the sense that the one-arm and 2 j-alternating-arm critical exponents (and therefore also the connectivity and volume exponents) are constant across the family of such percolation processes. The isoradial graphs in question are those that satisfy certain weak conditions on their embedding and on their track system. This class of graphs includes, for example, isoradial embeddings of periodic graphs, and graphs derived from rhombic Penrose tilings.

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...