Bond percolation on isoradial graphs: criticality and universality

Grimmett, Geoffrey; Manolescu, Ioan

doi:10.1007/s00440-013-0507-y

Cited by 35 publications

(63 citation statements)

References 39 publications

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“…For this reason, in some works (e.g. [24]) the denomination box crossing property is used. The strong RSW property was known for Bernoulli percolation on the regular square lattice from the works of Russo and Seymour and Welsh [35,36], hence the name.…”

Section: Results For the Classical Random-cluster Modelmentioning

confidence: 99%

“…We will not discuss this generalisation here and simply stick to the case of doubly-periodic graphs. Interested readers may consult [24] for the exact conditions required for G; the proofs below adapt readily.…”

Section: Results For the Classical Random-cluster Modelmentioning

confidence: 99%

“…In this section we will consider isoradial embeddings of the square lattice. As described in [24], a procedure based on track exchanges transforms one isoradial embedding of the square lattice into a different one. In addition to [24], the effect of boundary conditions needs to be taken into account; a construction called convexification is therefore required.…”

Section: From Regular Square Lattice To Isoradial Square Latticementioning

confidence: 99%

“…In [24], the gray rhombus was added before exchangin the tracks and removed afterwards. Thus, the track exchange could be perceived as a measure-and connection-preserving transformation between isoradial square lattices.…”

Section: Track Exchangementioning

confidence: 99%

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Universality for the random-cluster model on isoradial graphs

Duminil-Copin¹,

Li²,

Manolescu³

2018

Electron. J. Probab.

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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...

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Section: Results For the Classical Random-cluster Modelmentioning

confidence: 99%

Section: Results For the Classical Random-cluster Modelmentioning

confidence: 99%

Section: From Regular Square Lattice To Isoradial Square Latticementioning

confidence: 99%

Section: Track Exchangementioning

confidence: 99%

See 2 more Smart Citations

Universality for the random-cluster model on isoradial graphs

Duminil-Copin¹,

Li²,

Manolescu³

2018

Electron. J. Probab.

Self Cite

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“…2b). We note that another approach to finding thresholds for some lattices has recently been put forth by Grimmett and Manolescu [16], for lattices that can be represented in an isoradial form in which each polygon can be inscribed in a circle of equal radius. This method can be used to find a geometrical proof [17] of Wu's criticality condition for the square checkerboard lattice [18], however, it cannot be used to study the types of square lattices considered here.…”

Section: Introductionmentioning

confidence: 99%

Percolation on hypergraphs with four-edges

Damavandi

Ziff

2015

J. Phys. A: Math. Theor.

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Abstract. We study percolation on self-dual hypergraphs that contain hyperedges with four bounding vertices, or "four-edges", using three different generators, each containing bonds or sites with three distinct probabilities p, r, and t connecting the four vertices. We find explicit values of these probabilities that satisfy the selfduality conditions discussed by Bollobás and Riordan. This demonstrates that explicit solutions of the self-duality conditions can be found using generators containing bonds and sites with independent probabilities. These solutions also provide new examples of lattices where exact percolation critical points are known. One of the generators exhibits three distinct criticality solutions (p, r, t). We carry out Monte-Carlo simulations of two of the generators on two different hypergraphs to confirm the critical values. For the case of the hypergraph and uniform generator studied by Wierman et al., we also determine the threshold p = 0.441374 ± 0.000001, which falls within the tight bounds that they derived. Furthermore, we consider a generator in which all or none of the vertices can connect, and find a soluble inhomogeneous percolation system that interpolates between site percolation on the union-jack lattice and bond percolation on the square lattice.arXiv:1506.06125v2 [cond-mat.dis-nn]

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Isoradial immersions

Boutillier

Cimasoni

Tilière

2021

Journal of Graph Theory

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Isoradial embeddings of planar graphs play a crucial role in the study of several models of statistical mechanics, such as the Ising and dimer models. Kenyon and Schlenker give a combinatorial characterization of planar graphs admitting an isoradial embedding, and describe the space of such embeddings. In this paper we prove two results of the same type for generalizations of isoradial embeddings: isoradial immersions and minimal immersions. We show that a planar graph admits a flat isoradial immersion if and only if its train-tracks do not form closed loops, and that a bipartite graph has a minimal immersion if and only if it is minimal. In both cases we describe the space of such immersions. The techniques used are different in both settings, and distinct from those of Kenyon and Schlenker. We also give an application of our results to the dimer model defined on bipartite graphs admitting minimal immersions.

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Bond percolation on isoradial graphs: criticality and universality

Cited by 35 publications

References 39 publications

Universality for the random-cluster model on isoradial graphs

Universality for the random-cluster model on isoradial graphs

Percolation on hypergraphs with four-edges

Isoradial immersions

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