The Random-Cluster Model

Grimmett, Geoffrey

doi:10.1007/978-3-540-32891-9

Cited by 368 publications

(791 citation statements)

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“…We briefly describe this connection in the case of the q-state Potts model on the square lattice. More details and references are found in [She06], [Car07], [Gri06].…”

Section: Introductionmentioning

confidence: 99%

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Conformal Radii for Conformal Loop Ensembles

Schramm

Sheffield⋆

Wilson

2009

Commun. Math. Phys.

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The conformal loop ensembles CLE κ , defined for 8/3 ≤ κ ≤ 8, are random collections of loops in a planar domain which are conjectured scaling limits of the O(n) loop models. We calculate the distribution of the conformal radii of the nested loops surrounding a deterministic point. Our results agree with predictions made by Cardy and Ziff and by Kenyon and Wilson for the O(n) model. We also compute the expectation dimension of the CLE κ gasket, which consists of points not surrounded by any loop, to bewhich agrees with the fractal dimension given by Duplantier for the O(n) model gasket.

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“…We briefly describe this connection in the case of the q-state Potts model on the square lattice. More details and references are found in [She06], [Car07], [Gri06].…”

Section: Introductionmentioning

confidence: 99%

“…(With this choice of p, the FK model is self-dual and believed to be critical, see e.g., [Gri06,Chapter 6]. )…”

Section: Introductionmentioning

confidence: 99%

Conformal Radii for Conformal Loop Ensembles

Schramm

Sheffield⋆

Wilson

2009

Commun. Math. Phys.

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“…It is standard (as in [12,22]) that we may find ν such that ω 1 and ω 2 are identical configurations within the region of Λ m,β that is not connected to ∂ h Λ m,β in the upper configuration ω 2 . Let D be the set of all pairs (ω 1 , ω 2 ) ∈ Ω n,β × Ω n,β such that: ω 2 contains no path joining ∂B to ∂ h Λ m,β , where…”

Section: And All Admissible Random-cluster Boundary-conditions τ and mentioning

confidence: 96%

“…See [19,20] for details of the contact model. Just as the percolation model on a lattice may be generalised to the socalled random-cluster model (see [12]), so may the continuum percolation model be extended to a continuum random-cluster model. We shall work here mostly on a bounded box rather than the whole space Z × R. Let a, b ∈ Z, s, t ∈ R satisfy a ≤ b, s ≤ t, and write Λ = [a, b] × [s, t] for the box {a, a + 1, .…”

Section: The Continuum Random-cluster Modelmentioning

confidence: 99%

Entanglement in the Quantum Ising Model

2008

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We study the asymptotic scaling of the entanglement of a block of spins for the ground state of the one-dimensional quantum Ising model with transverse field. When the field is sufficiently strong, the entanglement grows at most logarithmically in the number of spins. The proof utilises a transformation to a model of classical probability called the continuum random-cluster model, and is based on a property of the latter model termed ratio weak-mixing. Our proof applies equally to a large class of disordered interactions. The quantum Ising modelThe quantum Ising model in a transverse magnetic field is one of the most famous examples of exactly solvable one-dimensional quantum models. The solution was first given by Pfeuty in [26], based on earlier works by Lieb, Schultz, and Mattis [18] and by McCoy [21]. The diagonalisation of the Hamiltonian and the determination of the energy eigenstates is based on methods developed by Jordan and Wigner [16] in the theory of second quantisation of fermion fields, and by Bogoliubov [7] in the theory of superconductivity. This model exhibits a second-order phase transition in the ground state when the temperature of the system is zero. The existence of the phase transition and

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“…Stochastic domination and the FKG inequality -dealing with partial ordering of spin configurations, functions thereof and thus also measures -are discussed in, e.g., Georgii [57] or Grimmett [61]. The proof of (2.22) can alternatively be based on Griffiths' correlation inequalities (Griffiths [60]).…”

Section: Literature Remarksmentioning

confidence: 99%