A power-law upper bound on the correlations in the 2D random field Ising model

TL;DR

This paper establishes a power-law upper bound on the decay of boundary effects on magnetization in the 2D random field Ising model, valid for all temperatures and field strengths, including zero, extending previous exponential decay results.

Contribution

It provides the first power-law upper bound on boundary influence decay in the 2D RFIM for all parameters, generalizing prior results limited to strong disorder or high temperature.

Findings

Power-law decay of boundary effects at all temperatures.

Bound holds for all field strengths, including zero.

Analysis extends the Aizenman-Wehr proof to quantify decay rates.

Abstract

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.