Typical Gibbs configurations for the 1d Random Field Ising Model with long range interaction

TL;DR

This paper analyzes the typical configurations of a one-dimensional long-range Ising model with random fields, revealing phase behavior and typical interval lengths depending on the decay parameter and randomness strength.

Contribution

It provides a detailed characterization of typical spin configurations in the 1D RFIM with long-range interactions, including phase structure and interval length estimates.

Findings

Typical configurations are intervals of aligned spins with lengths depending on parameters.

Phase behavior changes with the decay parameter  and randomness variance.

Interval lengths grow as a power law or exponentially depending on .

Abstract

We study a one--dimensional Ising spin systems with ferromagnetic, long--range interaction decaying as $n^{-2+\a}$, $\a \in [0,\frac 12]$, 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, gaussian or subgaussian with variance $\theta$. We show that for temperature and variance of the randomness small enough, with an overwhelming probability with respect to the random fields, the typical configurations, within volumes centered at the origin whose size grow faster than any power of $\th^{-1}$, % {\bf around the origin} are intervals of $+$ spins followed by intervals of $-$ spins whose typical length is $ \simeq \th^{-\frac{2}{(1-2\a)}}$ for $0\le \a<1/2$ and $\simeq e^{\frac 1 {\th^{2}}}$ for $\a=1/2$.