Exponential decay of correlations in the two-dimensional random field Ising model

Ding, Jian; Xia, Jiaming

doi:10.1007/s00222-020-01024-y

Cited by 20 publications

(24 citation statements)

References 16 publications

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“…Classic results include the work of Aizenman and Wehr [AW90], who showed that in d " 2, as long as there is randomness in the external field, the boundary influence always decays to 0 as N Ñ 8. Recently, this decay was shown to be exponential by Ding and Xia [DX21] and also Aizenman, Harel and Peled [AHP20] for Gaussian random fields. In contrast, in d ě 3, Bricmont and Kupiainen [BK88] showed that when the distribution of the random field ω v has a sufficiently fast decaying Gaussian tail, then at sufficiently low temperatures, the boundary influence does not vanish as N Ñ 8.…”

Section: Introductionmentioning

confidence: 89%

A New Correlation Inequality for Ising Models with External Fields

Ding¹,

Song²,

Sun³

2021

Preprint

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We study ferromagnetic Ising models on finite graphs with an inhomogeneous external field, where a subset of vertices is designated as the boundary. We show that the influence of boundary conditions on any given spin is maximised when the external field is identically 0. One corollary is that spin-spin correlation is maximised when the external field vanishes. In particular, the random field Ising model on Z d , d ě 3, exhibits exponential decay of correlations in the entire high temperature regime of the pure Ising model. Another corollary is that the pure Ising model in d ě 3 satisfies the conjectured strong spatial mixing property in the entire high temperature regime.

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Section: Introductionmentioning

confidence: 89%

A New Correlation Inequality for Ising Models with External Fields

Ding¹,

Song²,

Sun³

2021

Preprint

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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%

“…A∈A (q − 1) h∈E σ:Aσ=A ν 1 (h, σ)dh q−1 j=1 h∈E σ:Aσ =A ν 1 (h A,j , σ A,j )dh + P(E c ) (since h σ ν 1 (h A,j , σ A,j )dh = 1) A∈A (q − 1)e − |∂A| 2(q−1)T + P(E c ) (by (26) and by definition of E) e −c/T + e −c/ε 2 (by (15) and Lemma 4.2) , as required.…”

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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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“…The surprising phenomena is that this picture changes for the random field Ising model: on Z d for d ≥ 3 there is still a phase transition if the variance of the random fields h x is not too large [7], but on Z 2 there is no phase transition if the variance is non-zero [1]. In fact, in recent breakthroughs, it was shown that on Z 2 correlations always decay exponentially [13], and on Z d , d ≥ 3, correlations decay exponentially throughout the high-temperature regime [12]. It is natural to wonder if there are algorithmic counterparts to these physical phenomena.…”

Section: Introductionmentioning

confidence: 99%

Approximation algorithms for the random-field Ising model

Helmuth¹,

Lee²,

Perkins³

et al. 2021

Preprint

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Approximating the partition function of the ferromagnetic Ising model with general external fields is known to be #BIS-hard in the worst case, even for bounded-degree graphs, and it is widely believed that no polynomial-time approximation scheme exists. This motivates an average-case question: are there classes of instances for which polynomial-time approximation schemes exist? We investigate this question for the random field Ising model on graphs with maximum degree ∆. We establish the existence of fully polynomial-time approximation schemes and samplers with high probability over the random fields if the external fields are IID Gaussians with variance larger than a constant depending only on the inverse temperature and ∆. The main challenge comes from the positive density of vertices at which the external field is small. These regions, which may have connected components of size Θ(log n), are a barrier to algorithms based on establishing a zero-free region, and cause worst-case analyses of Glauber dynamics to fail. The analysis of our algorithm is based on percolation on a self-avoiding walk tree.

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Exponential decay of correlations in the two-dimensional random field Ising model

Cited by 20 publications

References 16 publications

A New Correlation Inequality for Ising Models with External Fields

A New Correlation Inequality for Ising Models with External Fields

Long range order for random field Ising and Potts models

Approximation algorithms for the random-field Ising model

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