Correlation length of the two-dimensional random field Ising model via greedy lattice animal

Abstract

For the two-dimensional random field Ising model where the random field is given by i.i.d.\ mean zero Gaussian variables with variance $\hat{I}{\mu }^2$, we study (one natural notion of) the correlation length, which is the critical size of a box at which the influences of the random field and of the boundary condition on the spin magnetization are comparable. We show that as $\hat{I}\mu \to 0$, at zero temperature the correlation length scales as $e^{\hat{I}˜(\hat{I}{\mu }^{-4/3})}$ (and our upper bound applies for all positive temperatures).

Section 4 explains (without claiming credit) the proof of greedy lattice animal using Talagrand (2014); Section 5 keeps our `original' proof as a record (which will be omitted from the journal version)