On the decay of correlations inSO(n)-symmetric ferromagnets

McBryan, Oliver A.; Spencer, Thomas

doi:10.1007/bf01609854

Cited by 122 publications

(72 citation statements)

References 5 publications

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“…These results are similar to the power law obtained in [MS] for the 2D XY model. The proofs in [MR1,MR2,DMR,MS] use a deformation of the statistical mechanics measure and is related to Mermin-Wagner.…”

Section: Remarksmentioning

confidence: 99%

Duality, statistical mechanics and random matrices

Spencer

2012

Current Developments in Mathematics

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Abstract. This article will present an informal review of some results and conjectures about the spectral theory of large random matrices and related spin systems in statistical mechanics. A class of lattice spin models provides a dual representation for spectral problems in random matrix theory. Ordered and disordered phases of the spins correspond to different spectral types and quantum time evolutions. In three dimensions, we describe a phase transition for a supersymmetric statistical mechanics system inspired by random matrix theory. This transition has a classical interpretation in terms of a history dependent walk on the lattice. In the ordered phase the walk is diffusive while in the disordered phase it is localized near its starting point.

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Section: Remarksmentioning

confidence: 99%

Duality, statistical mechanics and random matrices

Spencer

2012

Current Developments in Mathematics

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“…[69,70,71]. In particular, magnetic and nematic long-range order are impossible, and all correlation functions decay at least as an inverse power of the distance [72,53].…”

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confidence: 99%

Pathologies of the large-N limit for RPN−1, CPN−1, QPN−1 and mixed isovector/isotensor σ-models

Sokal¹,

Starinets²

2001

Nuclear Physics B

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We compute the phase diagram in the N → ∞ limit for lattice RP N −1 , CP N −1 and QP N −1 σ-models with the quartic action, and more generally for mixed isovector/isotensor models. We show that the N = ∞ limit exhibits phase transitions that are forbidden for any finite N . We clarify the origin of these pathologies by examining the exact solution of the one-dimensional model: we find that there are complex zeros of the partition function that tend to the real axis as N → ∞. We conjecture the correct phase diagram for finite N as a function of the spatial dimension d. Along the way, we prove some new correlation inequalities for a class of N -component σ-models, and we obtain some new results concerning the complex zeros of confluent hypergeometric functions.

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“…The existence of such an a gives that the pair correlations of the O(n) model, n ≥ 2, decay exponentially, using the argument of McBryan and Spencer [12]. The proof in that paper carries through with no change.…”

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confidence: 88%

“…Let us sketch the argument of [12] as well (even though [12] is very short and to the point). It is a version of the Mermin-Wagner argument where quadratic control of the error is achieved using a complex change of variables.…”

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confidence: 99%

Continuous Versus Discrete Spins in the Hyperbolic Plane

Benjamini

Kozma

2017

J Stat Phys

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Abstract. We study the O(n) model on graphs quasi-isometric to the hyperbolic plane, with free boundary conditions. We observe that the pair correlation decays exponentially with distance, for all temperatures, if and only if n > 1.We wish to report here on a curious contrast between the behaviour of models with discrete and continuous symmetry on graphs quasi-isometric to the hyperbolic plane. For concreteness we will study the O(n) model, with n = 1, or Ising, being our model for a model with discrete spins and n ≥ 2 our model for continuous spins and symmetry group. Let us start with the relevant definitions.Definition. Two metric spaces X and Y are called quasi-isometric if there is a map φ : X → Y with the propertieswhere c and C denote constants which are independent of x and y.We say that a graph is quasi-isometric to the hyperbolic plane if, when you equip it with the graph metric (i.e. the distance between any two vertices is the length of the shortest path between them), it is quasi-isometric to the hyperbolic plane with its standard metric.An easy example of a graph quasi-isometric to the hyperbolic plane is the seven-regular planar triangulation: the (unique) graph which is the 1-skeleton of an infinite triangulation of a planar disk which satisfies that the degree of every vertex is 7 (the number 7 may be replaced with any n > 6 and the graph would still be quasi-isometric to the hyperbolic plane). This

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On the decay of correlations inSO(n)-symmetric ferromagnets

Cited by 122 publications

References 5 publications

Duality, statistical mechanics and random matrices

Duality, statistical mechanics and random matrices

Pathologies of the large-N limit for RPN−1, CPN−1, QPN−1 and mixed isovector/isotensor σ-models

Continuous Versus Discrete Spins in the Hyperbolic Plane

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