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

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 ${\epsilon }^2$. Our primary result is the following phase transition at $T=T_c$: for $\epsilon \ll N^{-7\mathrm{/}8}$ 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 $N^{-1\mathrm{/}8}$ and thus the disorder essentially has no effect on the boundary influence; for $\epsilon \gg N^{-7\mathrm{/}8}$, the boundary influence decays as $N^{-\frac{1}{8}}{e}^{-\mathrm{\Theta }({\epsilon }^{8\mathrm{/}7}N)}$ (i.e., the disorder contributes a factor of $e^{-\mathrm{\Theta }({\epsilon }^{8\mathrm{/}7}N)}$ to the decay rate). For a natural notion of the correlation length, i.e., 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 $\epsilon \downarrow 0$ the correlation length transits from $\mathrm{\Theta }({\epsilon }^{-8\mathrm{/}7})$ at $T_c$ to $e^{\mathrm{\Theta }({\epsilon }^{-4\mathrm{/}3})}$ for $T<T_c$.