The Ising magnetization exponent on $$\mathbb{Z }^2$$ is $$1/15$$

Camia, Federico; Garban, Christophe; Newman, Charles M.

doi:10.1007/s00440-013-0526-8

Cited by 11 publications

(17 citation statements)

References 23 publications

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“…Our final main result (in Theorem 4) gives very precise behavior for the magnetization M(H) (expected spin value of the (β c , H) Ising model on Z 2 ) that improves the bounds from [3] that as H ↓ 0…”

Section: Introductionsupporting

confidence: 54%

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FK–Ising coupling applied to near-critical planar models

Camia

Jian

Newman

2020

Stochastic Processes and their Applications

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We consider the Ising model at its critical temperature with external magnetic field ha 15/8 on aZ 2 . We give a purely probabilistic proof, using FK methods rather than reflection positivity, that for a = 1, the correlation length is ≥ const. h −8/15 as h ↓ 0. We extend to the a ↓ 0 continuum limit the FK-Ising coupling for all h > 0, and obtain tail estimates for the largest renormalized cluster area in a finite domain as well as an upper bound with exponent 1/8 for the one-arm event. Finally, we show that for a = 1, the average magnetization, M(h), in Z 2 satisfies M(h)/h 1/15 → some B ∈ (0, ∞) as h ↓ 0.

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Section: Introductionsupporting

confidence: 54%

“…(2) 1 The derivation of (1) in [3] was fairly short, but the derivation of (2) in Subsection 1.2 below is yet shorter and uses little more than the existence of a scaling limit magnetization field for h ≥ 0 [4,5].…”

Section: Introductionmentioning

confidence: 99%

FK–Ising coupling applied to near-critical planar models

Camia

Jian

Newman

2020

Stochastic Processes and their Applications

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“…The following is a consequence of Theorem 2 and the scaling properties of Φ h (as discussed in Subsection 1.1 and Section 6). This result complements that of [9] that the (H ↓ 0 at β c ) Ising magnetization exponent is 1/15:…”

Section: 2supporting

confidence: 86%

Exponential Decay for the Near‐Critical Scaling Limit of the Planar Ising Model

Camia

Jian

Newman

2020

Comm Pure Appl Math

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We consider the Ising model at its critical temperature with external magnetic field ha 15=8 on the square lattice with lattice spacing a. We show that the truncated two-point function in this model decays exponentially with a rate independent of a as a # 0. As a consequence, we show exponential decay in the near-critical scaling limit Euclidean magnetization field. For the lattice model with a D 1, the mass (inverse correlation length) is of order h 8=15 as h # 0;for the Euclidean field, it equals exactly C h 8=15 for some C . Although there has been much progress in the study of critical scaling limits, results on nearcritical models are far fewer due to the lack of conformal invariance away from the critical point. Our arguments combine lattice and continuum FK representations, including coupled conformal loop and measure ensembles, showing that such ensembles can be useful even in the study of near-critical scaling limits. Thus we provide the first substantial application of measure ensembles.

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“…Consider, e.g., the average magnetization σ 0 βc,h in the critical Ising model on Z 2 with a homogeneous external field h > 0. If relation (1.20) holds (with d = 2, λ δ =λ δ which would sharpen the results in [CGN12b]. Analogous predictions can be formulated for disorder pinning and directed polymer models (see Section 3).…”

Section: (Universality)mentioning

confidence: 72%

Polynomial chaos and scaling limits of disordered systems

Caravenna¹,

Sun²,

Zygouras³

2016

J. Eur. Math. Soc.

140

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Abstract. Inspired by recent work of Alberts, Khanin and Quastel [AKQ14a], we formulate general conditions ensuring that a sequence of multi-linear polynomials of independent random variables (called polynomial chaos expansions) converges to a limiting random variable, given by a Wiener chaos expansion over the d-dimensional white noise. A key ingredient in our approach is a Lindeberg principle for polynomial chaos expansions, which extends earlier work of Mossel, O'Donnell and Oleszkiewicz [MOO10]. These results provide a unified framework to study the continuum and weak disorder scaling limits of statistical mechanics systems that are disorder relevant, including the disordered pinning model, the (long-range) directed polymer model in dimension 1 + 1, and the two-dimensional random field Ising model. This gives a new perspective in the study of disorder relevance, and leads to interesting new continuum models that warrant further studies.

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The Ising magnetization exponent on $$\mathbb{Z }^2$$ is $$1/15$$

Cited by 11 publications

References 23 publications

FK–Ising coupling applied to near-critical planar models

FK–Ising coupling applied to near-critical planar models

Exponential Decay for the Near‐Critical Scaling Limit of the Planar Ising Model

Polynomial chaos and scaling limits of disordered systems

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