A Note on Exponential Decay in the Random Field Ising Model

Abstract

For the two-dimensional random field Ising model (RFIM) with bimodal (i.e., two-valued) external field, we prove exponential decay of correlations either (i) when the temperature is larger than the critical temperature of the Ising model without external field and the magnetic field strength is small or (ii) at any temperature when the magnetic field strength is sufficiently large. Unlike previous work on exponential decay, our approach is not based on cluster expansions but rather on arguably simpler methods; these combine an analysis of the Kertész line and a coupling of Ising measures (and also their random cluster representations) with different boundary conditions. We also show similar but weaker results for the RFIM with a general field distribution and in any dimension. neither g + nor g − , h e := h z for each external edge e = {z, g + } or {z, g − }, 1 {·} denotes the indicator function, and other notation is similar to that in (19). We now define P

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

“…Results in this vein for systems related to the RFIM can be found in the works of A. Berretti [6], J. Imbrie and J. Fröhlich [16], and F. Camia, J. Jiang and C.M. Newman [9].…”

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

“…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]. The more interesting question is the nature of the decay at weak disorder, and in particular, whether there is a qualitative difference between weak and very weak disorder, with transition from exponential decay at weak disorder to power-law decay at very weak disorder [16,14,9].…”

Approximation algorithms for the random-field Ising model

“…In Section 4, we prove Theorem 3, the extension of our main result to random graphs. Finally, in Section 5, we show that our work generalizes one of the main theorems of [10] to infinite graphs of max-degree ∆. In Appendix A, we give details of the SAW tree construction and recursion used by the algorithms, and in Appendix B, we write out the algorithms explicitly.…”

“…The key mechanism behind the proof of Theorem 2 is that large external fields cause the system to rapidly decorrelate, as spins tend to align with their external field. We formalize this decorrelation by generalizing a disagreement percolation argument due to Camia, Jiang, and Newman [10]. The resulting notion of correlation decay is similar to, but somewhat weaker than, strong spatial mixing.…”