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 ${\mathit{\epsilon }}^2$, 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 $\mathit{\epsilon }\to 0$, at zero temperature the correlation length scales as $e^{\mathrm{\Theta }({\mathit{\epsilon }}^{-4∕3})}$ (and our upper bound applies for all positive temperatures).