## 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} := <φ, 1_{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) |
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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 2016 |

## Keywords

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

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty