On Long Range Ising Models with Random Boundary Conditions

TL;DR

This paper studies one-dimensional long-range Ising models with random boundary conditions, revealing how the nature of the metastate varies with the decay rate of interactions, especially distinguishing between different regimes of decay.

Contribution

It introduces a detailed analysis of metastates in long-range Ising models with random boundaries, highlighting new behaviors for intermediate decay rates.

Findings

Metastates are highly dispersed for decay rate $ ext{} extlessrac{1}{2}$.

For decay rate $ extgreaterrac{1}{2}$, metastates support only extremal Gibbs measures.

The behavior for decay rate exactly $rac{1}{2}$ remains an open question.

Abstract

We consider polynomial long-range Ising models in one dimension, with ferromagnetic pair interactions decaying with power $2-\alpha$ (for $0 \leq \alpha < 1$), and prepared with randomly chosen boundary conditions. We show that at low temperatures in the thermodynamic limit the finite-volume Gibbs measures do not converge, but have a distributional limit, the so-called metastate. We find that there is a distinction between the values of $\alpha$ less than or larger than $\frac{1}{2}$. For moderate, or intermediate, decay $\alpha < \frac{1}{2}$, the metastate is very dispersed and supported on the set of all Gibbs measures, both extremal and non-extremal, whereas for slow decays $\alpha > \frac{1}{2}$ the metastate is still dispersed, but has its support just on the set of the two extremal Gibbs measures, the plus measure and the minus measure. The former, moderate decays case, appears to be new and is due to the occurrence of almost sure boundedness of the random variable which is the sum of all interaction (free) energies between random and ordered half-lines, when the decay is fast enough, but still slow enough to get a phase transition ($\alpha>0$); while the latter, slow decays case, is more reminiscent of and similar to the behaviour of higher-dimensional nearest-neighbour Ising models with diverging boundary (free) energies. We leave the threshold case $\alpha=\frac{1}{2}$ for further studies.