A phase transition and critical phenomenon for the two-dimensional random field Ising model

Abstract

We study the random field Ising model in a two-dimensional box with side length $N$, where the external field is given by independent normal variables with mean $0$ and variance ${ϵ}^2$. Our primary result is the following phase transition at $T=T_c$: for $ϵ\ll N^{-7/8}$ the boundary influence (i.e., the difference between the spin averages at the center of the box with the plus and the minus boundary conditions) decays as $N^{-1/8}$ and thus the disorder essentially has no effect on the boundary influence; for $ϵ\gg N^{-7/8}$, the boundary influence decays as $N^{-1/8}{e}^{-\Theta ({ϵ}^{8/7}N)}$ (i.e., the disorder contributes a factor of $e^{-\Theta ({ϵ}^{8/7}N)}$ to the decay rate). For a natural notion of correlation length, namely the minimal size of the box where the boundary influence shrinks by a factor of $2$ from that with no external field, we also prove the following: as $ϵ\downarrow 0$ the correlation length transitions from $\Theta ({ϵ}^{-8/7})$ at $T_c$ to $e^{\Theta ({ϵ}^{-4/3})}$ for $T<T_c$.