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 -