A note on exponential decay in the random field Ising model

TL;DR

This paper proves exponential decay of correlations in the 2D random field Ising model under certain conditions, using novel methods that do not rely on cluster expansions, and extends results to more general fields and dimensions.

Contribution

It introduces a new approach to proving exponential decay in RFIM, analyzing the Kertész line and coupling measures without cluster expansions, and extends results to general fields and higher dimensions.

Findings

Exponential decay when temperature is above critical and field is small.

Exponential decay at any temperature with sufficiently large magnetic field.

Similar weaker results for general field distributions and in any dimension.

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 (1) when the temperature is larger than the critical temperature of the Ising model without external field and the magnetic field strength is small or (2) 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 Kert\'{e}sz 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.