TY - JOUR

T1 - A Power-Law Upper Bound on the Correlations in the 2D Random Field Ising Model

AU - Aizenman, Michael

AU - Peled, Ron

N1 - Publisher Copyright:
© 2019, The Author(s).

PY - 2019/12/1

Y1 - 2019/12/1

N2 - As first asserted by Y. Imry and S-K Ma, the famed discontinuity of the magnetization as function of the magnetic field in the two dimensional Ising model is eliminated, for all temperatures, through the addition of quenched random magnetic field of uniform variance, even if that is small. This statement is quantified here by a power-law upper bound on the decay rate of the effect of boundary conditions on the magnetization in finite systems, as function of the distance to the boundary. Unlike exponential decay which is only proven for strong disorder or high temperature, the power-law upper bound is established here for all field strengths and at all temperatures, including zero, for the case of independent Gaussian random field. Our analysis proceeds through a streamlined and quantified version of the Aizenman–Wehr proof of the Imry–Ma rounding effect.

AB - As first asserted by Y. Imry and S-K Ma, the famed discontinuity of the magnetization as function of the magnetic field in the two dimensional Ising model is eliminated, for all temperatures, through the addition of quenched random magnetic field of uniform variance, even if that is small. This statement is quantified here by a power-law upper bound on the decay rate of the effect of boundary conditions on the magnetization in finite systems, as function of the distance to the boundary. Unlike exponential decay which is only proven for strong disorder or high temperature, the power-law upper bound is established here for all field strengths and at all temperatures, including zero, for the case of independent Gaussian random field. Our analysis proceeds through a streamlined and quantified version of the Aizenman–Wehr proof of the Imry–Ma rounding effect.

UR - http://www.scopus.com/inward/record.url?scp=85067231282&partnerID=8YFLogxK

U2 - 10.1007/s00220-019-03450-3

DO - 10.1007/s00220-019-03450-3

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AN - SCOPUS:85067231282

SN - 0010-3616

VL - 372

SP - 865

EP - 892

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 3

ER -