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

In [CGN12], we proved that the renormalized critical Ising magnetization \Φᵃ:= a15/8 \∑x\∈ a\\, \² \σₓ \\, \δₓ converge a\→ 0 to a random distribution that we denoted by \Φ^\∞. The of this paper is to establish some fundamental properties satisfied by \Φ^\∞ and the near-critical fields \Φ\∞,h. More, we obtain the following results. \ [(i)] If A\⊂ \ is a bounded domain and if m=mA := <\Φ^\∞, 1A denotes the rescaled magnetization in A, then there is a constant c=cA>0 such {equation*} \log \{m > x} \{x\→ \∞}{\~} -c \\;¹⁶\\,.{equation*} In particular, this provides an alternative proof that the \Φ^\∞ is non-Gaussian (another proof of this fact would use the-point correlation functions established in \{CHI} which do not satisfy's formula). [(ii)] The random variable m=mA has a smooth {\ density} one has more precisely the following bound on its Fourier transform:|\\,t m |\≤ e^- ̃\\, |t|16/15. [(iii)] There exists a-parameter family \Φ\∞,h of near-critical scaling limits for the field in the plane with vanishingly small external magnetic. \