Abstract: In 1975, Imry and Ma argued that in the presence of a random field, the two-dimensional Ising model does not exhibit long-range order at any temperature, while in dimensions three and above long range order remains in the weak field and low temperature regime.
In the first part of the talk I will review the development of a rigorous proof of this prediction by several authors and in the second part I will present recent work with Jian Ding on the correlation length of the random field Ising model in two dimensions.