# Phase Transition in the 1d Random Field Ising Model with Long Range Interaction

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## Abstract

We study one–dimensional Ising spin systems with ferromagnetic, long–range interaction decaying as *n* ^{−2+α }, \({\alpha \in\left(\frac 12,\frac{\ln 3}{\ln2}-1\right)}\), 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, gaussian or subgaussian. We show, for temperature and strength of the randomness (variance) small enough, with *IP* = 1 with respect to the random fields, that there are at least two distinct extremal Gibbs measures.

## Keywords

Long Range Gibbs Measure Range Interaction Gibbs State Interface Point
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## References

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