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

In [CGN12], we proved that the renormalized critical Ising magnetization fields $\Phi^a:= a^{15/8} \sum_{x\in a\, \Z^2} \sigma_x \, \delta_x$ converge as $a\to 0$ to a random distribution that we denoted by $\Phi^\infty$. The purpose of this paper is to establish some fundamental properties satisfied by this $\Phi^\infty$ and the near-critical fields $\Phi^{\infty,h}$. More precisely, we obtain the following results. \bi [(i)] If $A\subset \C$ is a smooth bounded domain and if $m=m_A := <{\Phi^\infty, 1_A}$ denotes the limiting rescaled magnetization in $A$, then there is a constant $c=c_A>0$ such that {equation*} \log \Pb{m > x} \underset{x\to \infty}{\sim} -c \; x^{16}\,.{equation*} In particular, this provides an alternative proof that the field $\Phi^\infty$ is non-Gaussian (another proof of this fact would use the $n$-point correlation functions established in \cite{CHI} which do not satisfy Wick's formula). [(ii)] The random variable $m=m_A$ has a smooth {\it density} and one has more precisely the following bound on its Fourier transform: $|\Eb{e^{i\,t m}} |\le e^{- \tilde{c}\, |t|^{16/15}}$. [(iii)] There exists a one-parameter family $\Phi^{\infty,h}$ of near-critical scaling limits for the magnetization field in the plane with vanishingly small external magnetic field. \ei