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 independent and identically distributed mean zero Gaussian variables with variance ε2, 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 ε→0, at zero temperature the correlation length scales as eΘ(ε−4∕3) (and our upper bound applies for all positive temperatures).