TY - JOUR
T1 - Random Currents and Continuity of Ising Model’s Spontaneous Magnetization
AU - Aizenman, Michael
AU - Duminil-Copin, Hugo
AU - Sidoravicius, Vladas
N1 - Publisher Copyright:
© 2014, The Author(s).
PY - 2015/3
Y1 - 2015/3
N2 - 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 (Formula presented.) in d = 3 dimensions. The analysis also applies 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 the 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 lower dimensional cases. The exclusion applies to the one dimensional model with 1/r2 interaction for which the spontaneous magnetization is known to be discontinuous at Tc.
AB - 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 (Formula presented.) in d = 3 dimensions. The analysis also applies 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 the 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 lower dimensional cases. The exclusion applies to the one dimensional model with 1/r2 interaction for which the spontaneous magnetization is known to be discontinuous at Tc.
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U2 - 10.1007/s00220-014-2093-y
DO - 10.1007/s00220-014-2093-y
M3 - Article
AN - SCOPUS:84939887070
SN - 0010-3616
VL - 334
SP - 719
EP - 742
JO - Communications In Mathematical Physics
JF - Communications In Mathematical Physics
IS - 2
ER -