Phase Transition in the 1d Random Field ising model with long range interaction

TL;DR

This paper investigates a one-dimensional long-range Ising model with random fields, demonstrating the existence of multiple extremal Gibbs measures under certain conditions, indicating a phase transition.

Contribution

It establishes the occurrence of phase transitions in a 1D long-range Ising model with random fields, a phenomenon previously not well understood in such systems.

Findings

Multiple extremal Gibbs measures exist at low temperature and randomness.

Phase transition occurs due to long-range interactions and random fields.

Results hold for a range of decay parameters in the interaction.

Abstract

We study the one dimensional Ising model with ferromagnetic, long range interaction which decays as |i-j|^{-2+a}, 1/2< a<1, in the presence of an external random filed. we assume that the random field is given by a collection of independent identically distributed random variables, subgaussian with mean zero. We show that for temperature and strength of the randomness (variance) small enough with P=1 with respect to the distribution of the random fields there are at least two distinct extremal Gibbs measures.