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

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.

Exponential Decay of Correlations in the 2D Random Field Ising Model

“…Surface tension. We extend the past observations of [3] on the different relations of the surface tension with disagreement percolation. In particular, we present an upper bound on the surface tension between two surfaces in terms of the amount of the disagreement percolation flux through an arbitrary non-anticipatory random set separating the two.…”

“…Tortuosity of the disagreement paths. The improvement presented in this paper over [3] is based on the quantification of the observation, which was expressed in [3] at only a heuristic level, that the disagreement percolation is weakened by the fractality of its connected clusters. A path from this suggestion to a proof was pointed out, in a somewhat modified context, in the aforementioned work of Ding-Xia [15].…”

“…Use is made of the fact that, like in the case of other percolation models, fast enough power law decay implies exponential decay (cf. [3]). Anti-concentration bounds play a role in the analysis, as well as concentration bounds which control the contribution of exceptional random field configurations.…”

“…These relate to the decay of correlations among the local spin magnetizations, which are quasi-local functions of the quenched random field, as well as to the typical decay of spin correlations within the quenched state. As presented in [3] with d(u, v) the graph distance on Z 2 and 0 := (0, 0). It has been early recognized, and since then proven in a variety of ways, that at high disorder and/or high temperatures, m(L) decays exponentially fast [18,7,11].…”

“…Recent results on the subject include: i) an initial bound m(L) ≤ C(ε)/ √ log log L [12], ii) power-law bounds for all ε > 0 and T ≥ 0, including also all finite-range ferromagnetic interactions [3], and, more recently, iii) exponentially-decaying upper bounds for all ε > 0 [15], limited to the nearest-neighbor case and T = 0. The latter result answers (in the negative) the question of possible transition in ε for the RFIM's ground state, but stops short of addressing the additional effect of thermal fluctuations.…”