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+alpha), alpha is an element of [0, 1/2], 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 theta. 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 theta(-1), are intervals of + spins followed by intervals of spins whose typical length is similar or equal to theta-2/(1-2 alpha) for 0 <= a < 1/2 and between e 1/theta. and e 1/theta(2) for alpha = 1/2.