Typical Gibbs Configurations for the 1d Random Field Ising Model with Long Range Interaction

We study one–dimensional Ising spin systems with ferromagnetic, long–range interaction decaying as n ^?2+α , \({\alpha \in 0,\frac 12]}\), in the presence of external random fields. We assume that the random fields are given by a collection of symmetric, independent, identically distributed real random variables, which are gaussian or subgaussian with variance θ. We show that when the temperature and the variance of the randomness are sufficiently small, with overwhelming probability with respect to the random fields, the typical configurations, within intervals centered at the origin whose length grow faster than any power of θ ^?1, are intervals of + spins followed by intervals of ? spins whose typical length is \({ \simeq\,\theta^{-\frac{2}{(1-2\alpha)}}}\) for 0 ≤ α < 1/2 and between \({ e^{\frac{1}{\theta}}}\) and \({e^{\frac 1 {\theta^{2}}}}\) for α = 1/2.