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^{-7/8} 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 \epsilon \gg N^{-7/8} , the boundary influence decays as N^{- 1/8}e^{-\Theta(\epsilon^{8/7} N)} (i.e., the disorder contributes a factor of e^{-\Theta(\epsilon^{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 \epsilon\downarrow 0 the correlation length transitions from \Theta(\epsilon^{-8/7}) at T_{c} to e^{\Theta(\epsilon^{-4/3})} for T < T_{c} .