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

In (Ann. Probab. 43 (2015) 528.571), we proved that the renormalized critical Ising magnetization fields φ^a:= a^15/8 σ _x∈aℤ² 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 →}∼-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 = 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.