Planar Ising magnetization field II. Properties of the critical and near-critical scaling limits

Federico Camia, Christophe Garban, Charles Newman

Research output: Contribution to journalArticle

Abstract

In (Ann. Probab. 43 (2015) 528.571), we proved that the renormalized critical Ising magnetization fields φa:= a15/8 σ x∈aℤ2 Σx δx converge as a → 0 to a random distribution that we denoted by φ. The purpose of this paper is to establish some fundamental properties satisfied by this φ and the near-critical fields φ∞,h. More precisely, we obtain the following results. (i) If A ⊂ ℂ is a smooth bounded domain and if m = mA := A denotes the limiting rescaled magnetization in A, then there is a constant c = cA > 0 such that log ℙ[m > x]x →∼-cx16. In particular, this provides an alternative way of seeing that the field φ is non-Gaussian (another proof of this fact would use the explicit n-point correlation functions established in (Ann. Math. 181 (2015) 1087-1138) which do not satisfy Wick's formula). (ii) The random variable m = mA has a smooth density and one has more precisely the following bound on its Fourier transform: |E[eitm]| ≤ e-c|t|16/15. (iii) There exists a one-parameter family φ∞,h of near-critical scaling limits for the magnetization field in the plane with vanishingly small external magnetic field.

Original languageEnglish (US)
Pages (from-to)146-161
Number of pages16
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume52
Issue number1
DOIs
StatePublished - Feb 1 2016

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Scaling Limit
Ising
Magnetization
External Field
Correlation Function
Bounded Domain
Fourier transform
Limiting
Random variable
Magnetic Field
Denote
Converge
Scaling
Random variables
Magnetic field
Alternatives

Keywords

  • Conformal covariance
  • Ising magnetization field
  • Ising model
  • Near-criticality
  • Sub-Gaussian tails

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

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title = "Planar Ising magnetization field II. Properties of the critical and near-critical scaling limits",
abstract = "In (Ann. Probab. 43 (2015) 528.571), we proved that the renormalized critical Ising magnetization fields φa:= a15/8 σ x∈aℤ2 Σx δx converge as a → 0 to a random distribution that we denoted by φ. The purpose of this paper is to establish some fundamental properties satisfied by this φ and the near-critical fields φ∞,h. More precisely, we obtain the following results. (i) If A ⊂ ℂ is a smooth bounded domain and if m = mA := A denotes the limiting rescaled magnetization in A, then there is a constant c = cA > 0 such that log ℙ[m > x]x →∼-cx16. In particular, this provides an alternative way of seeing that the field φ is non-Gaussian (another proof of this fact would use the explicit n-point correlation functions established in (Ann. Math. 181 (2015) 1087-1138) which do not satisfy Wick's formula). (ii) The random variable m = mA has a smooth density and one has more precisely the following bound on its Fourier transform: |E[eitm]| ≤ e-c|t|16/15. (iii) There exists a one-parameter family φ∞,h of near-critical scaling limits for the magnetization field in the plane with vanishingly small external magnetic field.",
keywords = "Conformal covariance, Ising magnetization field, Ising model, Near-criticality, Sub-Gaussian tails",
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T1 - Planar Ising magnetization field II. Properties of the critical and near-critical scaling limits

AU - Camia, Federico

AU - Garban, Christophe

AU - Newman, Charles

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N2 - In (Ann. Probab. 43 (2015) 528.571), we proved that the renormalized critical Ising magnetization fields φa:= a15/8 σ x∈aℤ2 Σx δx converge as a → 0 to a random distribution that we denoted by φ. The purpose of this paper is to establish some fundamental properties satisfied by this φ and the near-critical fields φ∞,h. More precisely, we obtain the following results. (i) If A ⊂ ℂ is a smooth bounded domain and if m = mA := A denotes the limiting rescaled magnetization in A, then there is a constant c = cA > 0 such that log ℙ[m > x]x →∼-cx16. In particular, this provides an alternative way of seeing that the field φ is non-Gaussian (another proof of this fact would use the explicit n-point correlation functions established in (Ann. Math. 181 (2015) 1087-1138) which do not satisfy Wick's formula). (ii) The random variable m = mA has a smooth density and one has more precisely the following bound on its Fourier transform: |E[eitm]| ≤ e-c|t|16/15. (iii) There exists a one-parameter family φ∞,h of near-critical scaling limits for the magnetization field in the plane with vanishingly small external magnetic field.

AB - In (Ann. Probab. 43 (2015) 528.571), we proved that the renormalized critical Ising magnetization fields φa:= a15/8 σ x∈aℤ2 Σx δx converge as a → 0 to a random distribution that we denoted by φ. The purpose of this paper is to establish some fundamental properties satisfied by this φ and the near-critical fields φ∞,h. More precisely, we obtain the following results. (i) If A ⊂ ℂ is a smooth bounded domain and if m = mA := A denotes the limiting rescaled magnetization in A, then there is a constant c = cA > 0 such that log ℙ[m > x]x →∼-cx16. In particular, this provides an alternative way of seeing that the field φ is non-Gaussian (another proof of this fact would use the explicit n-point correlation functions established in (Ann. Math. 181 (2015) 1087-1138) which do not satisfy Wick's formula). (ii) The random variable m = mA has a smooth density and one has more precisely the following bound on its Fourier transform: |E[eitm]| ≤ e-c|t|16/15. (iii) There exists a one-parameter family φ∞,h of near-critical scaling limits for the magnetization field in the plane with vanishingly small external magnetic field.

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