Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Federico Camia, Christophe Garban, and Charles M. Newman

Full-text: Open access


In (Ann. Probab. 43 (2015) 528–571), we proved that the renormalized critical Ising magnetization fields $\varPhi^{a}:=a^{15/8}\sum_{x\in a\mathbb{Z}^{2}}\sigma_{x}\delta_{x}$ converge as $a\to0$ to a random distribution that we denoted by $\varPhi^{\infty}$. The purpose of this paper is to establish some fundamental properties satisfied by this $\varPhi^{\infty}$ and the near-critical fields $\varPhi^{\infty,h}$. More precisely, we obtain the following results.

(i) If $A\subset\mathbb{C}$ is a smooth bounded domain and if $m=m_{A}:=\langle\varPhi^{\infty},1_{A}\rangle$ denotes the limiting rescaled magnetization in $A$, then there is a constant $c=c_{A}>0$ such that

\[\log\mathbb{P}[m>x]\mathop{\sim}_{x\to\infty}-cx^{16}.\] In particular, this provides an alternative way of seeing that the field $\varPhi^{\infty}$ 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: $\vert \mathbb{E}[\mathrm{e}^{\mathrm{i}tm}]\vert \le \mathrm{e}^{-\tilde{c}\vert t\vert^{16/15}}$.

(iii) There exists a one-parameter family $\varPhi^{\infty,h}$ of near-critical scaling limits for the magnetization field in the plane with vanishingly small external magnetic field.


Dans l’article (Ann. Probab. 43 (2015) 528–571), nous avons montré que le champ de magnétisation du modèle d’Ising critique $\varPhi^{a}:=a^{15/8}\sum_{x\in a\mathbb{Z}^{2}}\sigma_{x}\delta_{x}$ converge lorsque $a\to0$ vers une distribution aléatoire limite $\varPhi^{\infty}$. Le but de cet article est d’analyser certaines propriétés fondamentales de cet objet limite $\varPhi^{\infty}$ ainsi que ses analogues presque-critiques $\varPhi^{\infty,h}$. Plus précisément, nous obtenons les résultats suivants :

(i) Si $A\subset\mathbb{C}$ est un domaine borné régulier du plan et si $m=m_{A}:=\langle\varPhi^{\infty},1_{A}\rangle$, alors il existe une constante $c=c_{A}>0$ telle que

\[\log\mathbb{P}[m>x]\mathop{\sim}_{x\to\infty}-cx^{16}.\] On obtient ainsi une preuve alternative du fait que $\varPhi^{\infty}$ est non-Gaussian (une autre façon de voir le coté non-Gaussien utilise les fonctions de corrélations à $n$-points obtenues dans (Ann. Math. 181 (2015) 1087–1138) qui ne satisfont pas la formule de Wick).

(ii) La variable aléatoire $m=m_{A}$ a une densité qui est analytique. Plus précisément, on obtient la borne suivante sur sa transformée de Fourier : $\vert \mathbb{E}[\mathrm{e}^{\mathrm{i}tm}]\vert \le \mathrm{e}^{-\tilde{c}\vert t\vert^{16/15}}$.

(iii) Il existe une famille à un paramètre $\varPhi^{\infty,h}$ de limite d’échelle presque-critiques pour le champ de magnétisation dans le plan avec un champ magnétique extérieur infinitésimal.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 1 (2016), 146-161.

Received: 1 October 2013
Revised: 29 August 2014
Accepted: 4 September 2014
First available in Project Euclid: 6 January 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B27: Critical phenomena 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

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


Camia, Federico; Garban, Christophe; Newman, Charles M. Planar Ising magnetization field II. Properties of the critical and near-critical scaling limits. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 1, 146--161. doi:10.1214/14-AIHP643.

Export citation


  • [1] D. Borthwick and S. Garibaldi. Did a 1-dimensional magnet detect a 248-dimensional Lie algebra? Notices Amer. Math. Soc. 58 (2011) 1055–1066.
  • [2] F. Camia, C. Garban and C. M. Newman. Planar Ising magnetization field I. Uniqueness of the critical scaling limit. Ann. Probab. 43 (2015) 528–571.
  • [3] F. Camia, C. Garban and C. M. Newman. The Ising magnetization exponent on $\mathbb{Z}^{2}$ is $1/15$. Probab. Theory Related Fields 160 (2014) 175–187.
  • [4] D. Chelkak and C. Hongler. In preparation.
  • [5] D. Chelkak, C. Hongler and K. Izyurov. Conformal invariance of spin correlations in the planar Ising model. Ann. of Math. (2) 181 (2015) 1087–1138.
  • [6] D. Chelkak, H. Duminil-Copin, C. Hongler, A. Kemppainen and S. Smirnov. Convergence of Ising interfaces to Schramm’s SLEs. C. R. Math. Acad. Sci. Paris Ser. I 352 (2014) 157–161.
  • [7] H. Duminil-Copin, C. Hongler and P. Nolin. Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model. Comm. Pure Appl. Math. 64 (2011) 1165–1198.
  • [8] C. Garban, G. Pete and O. Schramm. Pivotal, cluster and interface measures for critical planar percolation. J. Amer. Math. Soc. 26 (2013) 939–1024.
  • [9] R. B. Griffiths. Correlations in Ising ferromagnets. II. External magnetic fields. J. Math. Phys. 8 (1967) 484–489.
  • [10] R. B. Griffiths, C. A. Hurst and S. Sherman. Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys. 11 (1970) 790–795.
  • [11] Y. Kasahara. Tauberian theorems of exponential type. J. Math. Kyoto Univ. 18 (2) (1978) 209–219.
  • [12] B. M. McCoy and J.-M. Maillard. The importance of the Ising model. Progr. Theoret. Phys. 127 (2012) 791–817.
  • [13] B. M. McCoy and T. T. Wu. The Two-Dimensional Ising Model. Harvard Univ. Press, Cambridge, MA, 1973.
  • [14] M. Reed and B. Simon. Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness. Academic Press [Harcourt Brace Jovanovich, Publishers], New York–London, 1975.
  • [15] T. T. Wu. Theory of Toeplitz determinants and the spin correlations of the two-dimensional Ising model. I. Phys. Rev. 149 (1966) 380–401.