A phase transition for the two-dimensional random field Ising/FK-Ising model

Abstract

We study the total variation (TV) distance between the laws of the 2D Ising/FK-Ising model in a box of side-length $N$ with and without an i.i.d.\ Gaussian external field with variance $\hat{I}{\mu }^2$. Letting the external field strength $\hat{I}\mu =\hat{I}\mu (N)$ depend on the size of the box, we derive a phase transition for each model depending on the order of $\hat{I}\mu (N)$. For the random field Ising model, the critical order for $\hat{I}\mu$ is $N^{-1}$. For the random field FK-Ising model, the critical order depends on the temperature regime: for $T>T_c$, $T=T_c$ and $T\in (0,T_c)$ the critical order for $\hat{I}\mu$ is, respectively, $N^{-\frac{1}{2}}$, $N^{-\frac{15}{16}}$ and $N^{-1}$. In each case, as $N\to \infty$ the TV distance under consideration converges to $1$ when $\hat{I}\mu$ is above the respective critical order and converges to $0$ when below.