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

Abstract

In [CGN12], we proved that the renormalized critical Ising magnetization fields ${\Phi }^a:=a^{15/8}{\sum }_{x\in a{\mathbb{Z}}^2}{\sigma }_x{\delta }_x$ 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

arXiv comment18 pages, 1 figure