The spontaneous magnetization is proved to vanish continuously at the
critical temperature for a class of ferromagnetic Ising spin systems which
includes the nearest neighbor ferromagnetic Ising spin model on $Z^d$
in $d=3$ dimensions. The analysis applies also to higher dimensions, for which
the result is already known, and to systems with interactions of power law
decay. The proof employs in an essential way an extension of Ising model's
random current representation to the model's infinite volume limit. Using it,
we relate the continuity of the magnetization to the vanishing of the free
boundary condition Gibbs state's Long Range Order parameter. For reflection
positive models the resulting criterion for continuity may be established
through the infrared bound for all but the borderline case, of the one
dimensional model with $1/r^2$ interaction, for which the spontaneous
magnetization is known to be discontinuous at $T_c$.
Description
Random Currents and Continuity of Ising Model's Spontaneous Magnetization
%0 Journal Article
%1 aizenman2013random
%A Aizenman, Michael
%A Duminil-Copin, Hugo
%A Sidoravicius, Vladas
%D 2013
%K random representation walk
%R 10.1007/s00220-014-2093-y
%T Random Currents and Continuity of Ising Model's Spontaneous
Magnetization
%U http://arxiv.org/abs/1311.1937
%X The spontaneous magnetization is proved to vanish continuously at the
critical temperature for a class of ferromagnetic Ising spin systems which
includes the nearest neighbor ferromagnetic Ising spin model on $Z^d$
in $d=3$ dimensions. The analysis applies also to higher dimensions, for which
the result is already known, and to systems with interactions of power law
decay. The proof employs in an essential way an extension of Ising model's
random current representation to the model's infinite volume limit. Using it,
we relate the continuity of the magnetization to the vanishing of the free
boundary condition Gibbs state's Long Range Order parameter. For reflection
positive models the resulting criterion for continuity may be established
through the infrared bound for all but the borderline case, of the one
dimensional model with $1/r^2$ interaction, for which the spontaneous
magnetization is known to be discontinuous at $T_c$.
@article{aizenman2013random,
abstract = {The spontaneous magnetization is proved to vanish continuously at the
critical temperature for a class of ferromagnetic Ising spin systems which
includes the nearest neighbor ferromagnetic Ising spin model on $\mathbb Z^d$
in $d=3$ dimensions. The analysis applies also to higher dimensions, for which
the result is already known, and to systems with interactions of power law
decay. The proof employs in an essential way an extension of Ising model's
random current representation to the model's infinite volume limit. Using it,
we relate the continuity of the magnetization to the vanishing of the free
boundary condition Gibbs state's Long Range Order parameter. For reflection
positive models the resulting criterion for continuity may be established
through the infrared bound for all but the borderline case, of the one
dimensional model with $1/r^2$ interaction, for which the spontaneous
magnetization is known to be discontinuous at $T_c$.},
added-at = {2019-08-23T00:13:15.000+0200},
author = {Aizenman, Michael and Duminil-Copin, Hugo and Sidoravicius, Vladas},
biburl = {https://www.bibsonomy.org/bibtex/22d30583b74b7eefb6ef66f94833f79e7/gzhou},
description = {Random Currents and Continuity of Ising Model's Spontaneous Magnetization},
doi = {10.1007/s00220-014-2093-y},
interhash = {deb0ec7f0ee419d8d918f3ed469cae00},
intrahash = {2d30583b74b7eefb6ef66f94833f79e7},
keywords = {random representation walk},
note = {cite arxiv:1311.1937Comment: 33 pages, the last change amounted to an adjustment of the dimension of M_LRO},
timestamp = {2019-08-23T00:13:15.000+0200},
title = {Random Currents and Continuity of Ising Model's Spontaneous
Magnetization},
url = {http://arxiv.org/abs/1311.1937},
year = 2013
}