### Abstract

In (Ann. Probab. 43 (2015) 528.571), we proved that the renormalized critical Ising magnetization fields φ^{a}:= a^{15/8} σ _{x∈}aℤ^{2} Σ_{x} δ_{x} 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 = m_{A} := A denotes the limiting rescaled magnetization in A, then there is a constant c = c_{A} > 0 such that log ℙ[m > x]_{x →}∼-cx^{16}. 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 = m_{A} has a smooth density and one has more precisely the following bound on its Fourier transform: |E[e^{itm}]| ≤ 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 language | English (US) |
---|---|

Pages (from-to) | 146-161 |

Number of pages | 16 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 52 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2016 |

### Fingerprint

### 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

**Planar Ising magnetization field II. Properties of the critical and near-critical scaling limits.** / Camia, Federico; Garban, Christophe; Newman, Charles.

Research output: Contribution to journal › Article

*Annales de l'institut Henri Poincare (B) Probability and Statistics*, vol. 52, no. 1, pp. 146-161. https://doi.org/10.1214/14-AIHP643

}

TY - JOUR

T1 - Planar Ising magnetization field II. Properties of the critical and near-critical scaling limits

AU - Camia, Federico

AU - Garban, Christophe

AU - Newman, Charles

PY - 2016/2/1

Y1 - 2016/2/1

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.

KW - Conformal covariance

KW - Ising magnetization field

KW - Ising model

KW - Near-criticality

KW - Sub-Gaussian tails

UR - http://www.scopus.com/inward/record.url?scp=84958553700&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84958553700&partnerID=8YFLogxK

U2 - 10.1214/14-AIHP643

DO - 10.1214/14-AIHP643

M3 - Article

VL - 52

SP - 146

EP - 161

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

SN - 0246-0203

IS - 1

ER -