• Corpus ID: 207870485

# Gibbs Measures for Long-Range Ising Models

@article{Ny2019GibbsMF,
title={Gibbs Measures for Long-Range Ising Models},
author={Arnaud Le Ny},
journal={arXiv: Mathematical Physics},
year={2019}
}
• A. L. Ny
• Published 4 November 2019
• Physics
• arXiv: Mathematical Physics
This review-type paper is based on a talk given at the conference Etats de la Recherche en Mecanique statistique, which took place at IHP in Paris (December 10-14, 2018). We revisit old results from the 80's about one dimensional long-range polynomially decaying Ising models (often called Dyson models in dimension one) and describe more recent results about interface fluctuations and interface states in dimensions one and two.
2 Citations

## Figures from this paper

• Physics
Journal of Mathematical Physics
• 2022
We extend proofs of non-Gibbsianness of decimated Gibbs measures at low temperatures to include long-range as well as vector-spin interactions. Our main tools consist in a two-dimensional use of
• Mathematics
Journal of Statistical Physics
• 2020
We consider equilibrium states (that is, shift-invariant Gibbs measures) on the configuration space SZd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}

## References

SHOWING 1-10 OF 115 REFERENCES

• Mathematics
Ergodic Theory and Dynamical Systems
• 2017
We weaken the assumption of summable variations in a paper by Verbitskiy [On factors of $g$ -measures. Indag. Math. (N.S.) 22 (2011), 315–329] to a weaker condition, Berbee’s condition, in order for
• Physics
• 2005
Following Frohlich and Spencer, we study one dimensional Ising spin systems with ferromagnetic, long range interactions which decay as ∣x−y∣−2+α, 0⩽α⩽1∕2. We introduce a geometric description of the
• Mathematics
• 2001
We study the nearest-neighbour Ising model with a class of random boundary conditions, chosen from a symmetric i.i.d. distribution. We show for dimensions 4 and higher that almost surely the only
• Physics
• 1988
We rigorously establish the existence of an intermediate ordered phase in one-dimensional 1/|x−y|2 percolation, Ising and Potts models. The Ising model truncated two-point function has a power law
It is proved that these renormalized measures of the Ising model on $$\mathbb{Z}^{2}$$ are almost Gibbsian at any temperature and to analyse in detail its convex set of DLR measures.
For the 3-dimensional Ising model with long-range interaction, Gibbs states are constructed that are small perturbations of non-translation-invariant ground states. These ground states are in
We prove that all the translation invariant Gibbs states of the Ising model are a linear combination of the pure phases for any . This implies that the average magnetization is continuous for .
• Mathematics
• 2004
The infinite-volume limit behavior of the 2d Ising model under possibly strong random boundary conditions is studied. The model exhibits chaotic size-dependence at low temperatures and we prove that
We review what is known about the structure of the set of weak limiting states of the Ising and Potts models at low enough temperature, and in particular we prove that the mixture \frac{1}{2}(\mu
• Mathematics
• 2010
We study single-site stochastic and deterministic transforma- tions of one-dimensional Gibbs measures in the uniqueness regime with infinite-range interactions. We prove conservation of Gibbsianness