Exponential Decay of Correlations in the 2D Random Field Ising Model

Aizenman, Michael; Harel, Matan; Peled, Ron

doi:10.1007/s10955-019-02401-5

Cited by 23 publications

(31 citation statements)

References 20 publications

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“…The integer-valued Gaussian field (ZGF) over a connected planar graph G, of vertex set V and edge set E (with a locally-finite embedding in R 2 ), has as its basic variables the values of the random function n : V → Z. The ZGF finite-volume partition function in 2 , which by default is taken here with the Dirichlet boundary conditions, is…”

Section: The Zgf Model and Its Two Phasesmentioning

confidence: 99%

“…When G is doubly periodic 0 < λ c (G) < ∞. Furthermore, for the ZGF on Z 2 , λ c (Z 2 ) ≥ 2 3 ln 2, and at λ = 2 3 ln 2 the model is delocalized.…”

Section: The Zgf Model and Its Two Phasesmentioning

confidence: 99%

“…The graphs may be finite or infinite, the latter case conveniently viewed as the infinite-volume limit of its finite restrictions. Thus Z 2 is studied as a limit (L → ∞) of its truncated version 2 . By default we take the model with free boundary conditions.…”

Section: The Villain Modelmentioning

confidence: 99%

“…The bound stated in (1.1 1.1) is obtained through a summation of the above, in a method which applies quite generally to doubly-periodic planar graphs. To convey the argument in a relatively simple way we present it in the context of Z 2 . By (5.3 5.3) fast decay of spin correlations implies that in the corresponding ZGF long level lines occur only rarely.…”

Section: Lower Bounds On the Spin Correlation Functionmentioning

confidence: 99%

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Depinning in integer-restricted Gaussian Fields and BKT phases of two-component spin models

Aizenman¹,

Harel²,

Peled³

et al. 2021

Preprint

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It is shown that the Villain model of two-component spins over two dimensional lattices exhibits slow, non-summable, decay of correlations at any temperature at which the dual integer-valued Gaussian field exhibits depinning. For the latter, we extend the recent proof by P. Lammers of the existence of a depinning transition in the integervalued Gaussian field in two-dimensional graphs of degree three to all doubly-periodic graphs, in particular to Z 2 . Taken together these two statements yield a new perspective on the Berezinskii-Kosterlitz-Thouless phase transition in the Villain model, and complete a new proof of depinning in two-dimensional integer-valued height functions.

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Section: The Zgf Model and Its Two Phasesmentioning

confidence: 99%

“…When G is doubly periodic 0 < λ c (G) < ∞. Furthermore, for the ZGF on Z 2 , λ c (Z 2 ) ≥ 2 3 ln 2, and at λ = 2 3 ln 2 the model is delocalized.…”

Section: The Zgf Model and Its Two Phasesmentioning

confidence: 99%

Section: The Villain Modelmentioning

confidence: 99%

Section: Lower Bounds On the Spin Correlation Functionmentioning

confidence: 99%

See 2 more Smart Citations

Depinning in integer-restricted Gaussian Fields and BKT phases of two-component spin models

Aizenman¹,

Harel²,

Peled³

et al. 2021

Preprint

Self Cite

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“…There has been controversy over this prediction for quite some time, and it was finally proved to be correct by [20,8] for d = 3 and by [4] for d = 2. In recent works [11,3,15,2] quantitative bounds on the decay rate in dimension two were obtained, and in particular, exponential decay was finally established in [15,2]. In addition, the authors of [14] studied (a notion of) the correlation length, defined as…”

Section: Introductionmentioning

confidence: 99%

Long range order for random field Ising and Potts models

Ding,

Zhuang

2021

Preprint

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We present a new and simple proof for the classic results of Imbrie (1985) and Bricmont-Kupiainen (1988) that for the random field Ising model in dimension three and above there is long range order at low temperatures with presence of weak disorder. With the same method, we obtain a couple of new results: (1) we prove that long range order exists for the random field Potts model at low temperatures with presence of weak disorder in dimension three and above; (2) we obtain a lower bound on the correlation length for the random field Ising model at low temperatures in dimension two (which matches the upper bound in Ding-Wirth (2020)). Our proof is based on an extension of the Peierls argument with inputs from Chalker (1983), Fisher-Fröhlich-Spencer (1984, Ding-Wirth (2020) and Talagrand's majorizing measure theory (1980s) (and in particular, our proof does not involve the renormalization group theory).

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Long range order for random field Ising and Potts models

Ding

Zhuang

2023

Comm Pure Appl Math

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We present a new and simple proof for the classic results of Imbrie (1985) and Bricmont–Kupiainen (1988) that for the random field Ising model in dimension three and above there is long range order at low temperatures with presence of weak disorder. With the same method, we obtain a couple of new results: (1) we prove that long range order exists for the random field Potts model at low temperatures with presence of weak disorder in dimension three and above; (2) we obtain a lower bound on the correlation length for the random field Ising model at low temperatures in dimension two (which matches the upper bound in Ding–Wirth (2020)). Our proof is based on an extension of the Peierls argument with inputs from Chalker (1983), Fisher–Fröhlich–Spencer (1984), Ding–Wirth (2020) and Talagrand's majorizing measure theory (1980s) (and in particular, our proof does not involve the renormalization group theory).

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Exponential Decay of Correlations in the 2D Random Field Ising Model

Cited by 23 publications

References 20 publications

Depinning in integer-restricted Gaussian Fields and BKT phases of two-component spin models

Depinning in integer-restricted Gaussian Fields and BKT phases of two-component spin models

Long range order for random field Ising and Potts models

Long range order for random field Ising and Potts models

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